SET, MAPPING, BINARY OPERATION, GROUPS

Set: A well define collection of distinct objects is called set. Usually, the objects are called elements of the set

Null set or Empty set or Valid set: 

A set which has no member (or element) is called a null set. Null set is denoted by Φ or {  }.

Sub-sets: If every elements of a set A is an element of the set B, then A is said to be subset of B; symbolically we write A ⊆ B or B⊇ A.

If A ⊆ B and there is an element in B which is not in A, then A is said to be proper subset  of B denoted by A ⊂ B

If A is not a subset of B, we write A ⊄ B because b ∈ A but b ∉ B

NOTE:

i) If A ⊂B then x ∈ A => x ∈ B

ii) Every set is subset of itself i.e., A ⊂ A

iii) If  A ⊂B, then an element not in B is not in A.

iv) Null set is subset of every set. (Follow from (iii))

Equality of two sets: Two sets A and B are said to be equal if A ⊆ B and B ⊆ A; we write A= B.

Universal sets: In a context if every set is subset of another set, say U then U is called the universal set in that context.

Power Set: Let A be a set. Then the set of all sub-sets of A is called the power set of A. Power Set of A is denoted by P(A).

Remember Φ is subset of every set.

Note:

If A has n (finite) number of elements, then its power set contains 2ⁿ number of elements.

OPERATION ON SETS 

Union of two sets: Let A and B be two sets . The set consisting of all elements which belongs to A or to B or to both A and B is called the union of the sets A and B and is denoted by A∪B. Thus  if x ∈ A ∪ B then x belongs to atleast one of A and B .

We define in a similar way the union of n sets A₁, A₂, .....Aₙ

Intersection of two sets:  Let A and B be two sets. The set consisting of all elements which belong to both the sets A and B is called intersection of the sets A and B and is denoted by A ∩ B. Thus if x ∈ A ∩ B then x ∈ A and x ∈ B.

We define in a similar way the intersection of n sets A₁, A₂, .....Aₙ.

Disjoint sets: Two sets A and B are said to be disjoint if A∩ B = Φ

i.e., if they have no common elements.

Complement of a set: In a context, let  U be the universal set and A be other set (obviously then A ⊂ U). The set consisting of the elements which belongs to U but not to A is called compliment of A in U) symbolically it is written as Aᶜ, A' or bar A etc. thus x ∈ Aᶜ => x ∈ U but x ∉ A.

Difference of two sets: Let A and B be two Sets. Then the set consisting of all elements of A which do not belong to B is defined as A - B. Thus x∈ A - B => x ∈ A but x∉ B.

Symmetric difference of two sets: The set (A - B) ∪ (B - A) is called the symmetric difference of the two sets A and B. It is denoted by A ∆ B.

LAWS OF ALGEBRA OF SETS 

If A, B, C are three arbitrary sets and U is universal set then:

1) Idempotent Law: A ∪ A = A, A ∩ A = A

2) Commutative Law: A ∪ B = B ∪ A, A ∩ B = B ∩ A

3) Associative Law: A ∪ (B ∪ C)= (A ∪ B) ∪ C,

                                   A ∩ (B ∩ C) = (A ∩ B) ∩ C

4) Distributive Law: A ∪ (B ∩ C)= (A ∪ B) ∩ (A ∪ C)

                                   A ∩ (B ∪ C) = (A ∩  B)∪ (A∩ C)

5) Identity Law: A ∪ Φ = A; A ∩  U = A

                            A ∪ U = U ; A ∩ Φ = Φ

6) Complement Law: A ∪ A' = U ; A ∩ A' = Φ

                                     (A')'= A; (U)= Φ, Φ = U

7) D' Morgan Law: (A ∪ B)' = A' ∩ B' ; (A∩ B)' = A' ∪ B'

Cartesian Product: If A and B are two non-empty sets, we define the Cartesian product A x B to be the set of all ordered pairs whose first elements belongs to A and the second element belongs to B. That is 

A x B ={(x,y): x ∈A, y ∈ B}

Similarly A x B x C is {(x,y,z): x ∈A, y ∈ B,z ∈ C}

Note:

If R is the set of all real numbers, then R x R is the set of all ordered pair of real numbers and, as we know from co-ordinate geometry, is equivalent to the set of points in the Co-ordinate plane. Conventionally we write A² for A x A, A³ for Ax A x A, and so forth.

EXERCISE - A

1) If A= {a, 1,2,4,6,9} and B={a,2,b, 6,7,8} then A ∩B.           {a,2,6}

2) If A= {3,4,9,7,x,y}, B={4,x,7,8} and C={x,z,w} then A ∩ B ∩ C.           {x}

3) U={1,2,3,4,5,6,7,8,9,10} and A={6,1,7,8}, then find A' is.        {2,3,4,5,9,10}

4) If A={2,5,9,6,7,8}, B={1,2,6,0,a,b} then find 

a) A - B.                 {5,9,7,8}

b) B - A.                  {1,0,a,b}

5) If A={2,5,6}, B={1,9,5} then A ∆ B.           {2,6,1,9}

6) Write down all the subsets of the set S={p,q,r}. What is the power set of the set S.    

7) Let A={x: x be an integer and 1≤ x ≤ 10} and B={x : x be a prime integer less than 10}. Show that A ∩B.

8) Let Z be the set of integers and A, B, C, D are sub-sets of Z given by 

A = { x ∈Z/ 0 ≤ x ≤ 10}

B={x ∈ Z/5 ≤ x ≤ 15}

C={x ∈ Z/ x ≥ 5}

D={x ∈ Z/x ≤ 15}

Find 

a) A ∪B

b) A ∩B

c) B - C

d) A - D

9) If aN={ax: x ∈N} then find 4N ∩ 6N where N is set of all natural numbers.       12N

10) If A and B are two subsets of a set S, find (A ∩B) ∪ (A' ∩ B) in the simplest form, where A' denotes the complement of A.      

11) If A={3,4,5,6,7,9}, B={5,9,1,6} and C={3,2}, find all elements of the set (A ∆ B) x C where ∆= symmetric difference of two sets.         

12) Show that A ∪ (A' ∩B)= A ∪B.

13) If A, B, C are three sets such that A ∪B = A ∪ C and A ∩B = A ∩ C then show B= C.

14) Show that if A - B = A then A ∩ B =  Φ.

15) If Show that (X ∪ X') ∩ (X ∪ Φ)= X for a set X and null set Φ.

16) For three sets A, B, C if A∩ C= B∩C and A ∩C'= B ∩ C', then show that A= B (C' denotes the complement of C).

17) If A∩B = Φ, the then show that (B ∩A') ∪ B'= S where S is universal set.

18) If A, B, C are subsets of an universal set S, show that A x (B ∪C)= (A x B) ∪ (A x C).

19) If A, B, C are subsets of the universal set X, show that (A ∪B) ∩ (A ∩ B') ∪ (A∩ B) ∪ (A' ∩ B')= X.

20) If A ∪ B = A ∩ B show that A= B

21) U, V, W be three sets and U ⊂ V then show that U x W ⊂ V x W.

22) Show that (A - B) x C = (A x C) - (B x C).

RAW- A

MAPPINGS

A rule f that assigns to each element x in a set A a unique element f(x) in a set B, is called a mapping from A to B. It is written as f: A--> B.

If y= f(x) the element y in B is called the image of x and x is called the preimage of y. The set A is called domain and B is called the co-domain of f. The set of images taken by f, that is, the set {y ∈ B : y = f(x)}, is called the range of f.

Equality of two mappings: 

Two mappings f: A--> B and g: A--> B are said to be equal i.e., f= g for all x in A.

Example: The two mapping f(x)= x +1 and g(x)= (x⅖-1)/(x -1) are not equal since f: R--->R and g: R - {1}-->R.

One-one mapping (or Injective mapping).

A mapping f: A-->B is called one-one or Injective if f(x₁) = f(x₂) only when x₁ = x₂.

Example: f: R---> C defined by f(x) = x + I is one-one because f(x₁) = f(x₂)=>  x₁+ i = x₂+ i => x₁ = x₂.

Example: f: R---> R defined by f(x) = x² is not one-one because f(x₁) = f(x₂)= > x₁² = x₂² does not imply x₁. = x₂ (e.g.f(2)= f(-2) but 2≠ 2).

Example: f: R⁺ ---> R⁺ is set of all non negative reals) defined by f(x)= x¹ is one-one because f(x₁) = f(x₂)= > x₁² = x₂² => x₁. = x₂ (as x₁. = x₂ can not be negative).

Composite of two mappings:

Let f: A---B and g: B--> C be two mappings. Then the mapping gof: A---B> B defined by gof(x)= g{f(x)} is called composition of g with f.

Example: Let f: R---> C be defined by f(x)= x²+ 2i and g: C --->C be defined by g(x)= x + 3i then 

gof(x)= g{f(x)} = g(x²+ 2i)= x⅖+ 2i + 3i = x²+ 5i

Note that fog is not defined here.

Identity Mapping: A Mapping I: A---B>A defined by I(x)= x is called identity Mapping.

Inverse Mapping:

If f: A---B> is a bijective Mapping, there is a Mapping denoted by f⁻¹: B --> A such that  f⁻¹(y)= x, where y= f(x). This mapping  f⁻¹ is called inverse mapping of f.

Example: Consider the bijective Mapping f: R---R defined by f(x)= x +2. Then  f⁻¹: R---R>R is defined as f⁻¹(y)= y -2 or f⁻¹(x)= x -2.

Example: Consider the non- subjective mapping f: R---> C defined by f(x)= x + i. We see f⁻¹(x +2i) is undefined as x + 2i is not image of any element under f. So f⁻¹ does not exist for this mapping.

EXERCISE - A

1) Are two mappings f, g: R--->R defined by f(x)= (x³+ x²+ x +1)/(x²+1) and g(x)= x+1 equal?            Yes

2) Are two mappings f, g: R--->R defined by f(x)= (x³+ x)/x and g(x)= x²+1 equal?     N

3) Let f : R--->R defined by f(x)= (3x +4), x ∈ R (R is the set of all real numbers) show that f is one-one and onto.

4) Show that mapping f : N--->N defined by f(n)= (n+2), n ∈ N (N is the set of all natural numbers) is one-one but not onto.

5) Let R⁺ denote the set of all positive real numbers. Show that f : R⁺--->R⁺ defined by f(x)= x² + 1 ∀ R⁺ is not a surjective mapping.

6) A mapping f: R--->R is defined as follows:

f(x)= 1, if x is rational 

      = -1, if X is irrational 

R is the set of all real numbers. Find the range of f. What is f(√7)?

Is the mapping f one-one and onto ? Give reasons.

7) Show that mapping f : R--->R defined by f(x)= cosx, is neither one-one nor onto.

8) Show that mapping f : [-π/2,π/2] --->[-1,1] defined as f(x)= sinx is bijective.

9) Is the mapping f : Q--->Q defined by f(x)= x³, bijective? (Q is the set of all rational numbers).

10) Is the mapping f : R--->R defined by f(x)= x² bijective?  (R is the set of all real numbers) 

11) Let S be the set of all integers and f,g,h are three mappings from S to S defined by 

f(n)= n², g(n)= |n| and h(n)= n +1 for any n ∈S.

Show that 

i) only one-one among f,g,h is one-one

ii) fg = gf where (fg)(s)= f(g(s)), s ∈ S.

12) Find whether the mapping f: Z--Z defined by f(x)= |x - 3|, x ∈ Z, and Z is the set of all integers is onto or not? 

Justify your answer.

13) A mapping f : Z-->N is defined as follows:

f(n)= 2n for n ≥ 1

      = -2n +1 for all n ≤ 0.

Determine whether the mapping is one-to and onto.

14) Show that the mapping f : R--{3}-->R - {1} defined by f(x)= (x -2)/(x -3) is bijective?  

15) If f : R⁺--->R⁺ be defined by f(x)= x² +3, x ∈ R⁺ where R⁺ is the set of all positive real numbers, find f⁻¹.  

16) Two mappings If f : R--->R and g: R---> are defined by f(x)= x² +1; g(x)= x. Determine fog and gof (R is the set of all positive reals).

17) Two mappings If f : R--->R and g: R--->R(R is the set of real numbers) are defined by

 f(x)= x² ; g(x)= x - 1, x ≥ 0

                        = x, x < 0

Find the mappings fog and gof.

18) Let A={1,2,3}, B={p,q,r} and let f: A--->B defined by f(1)= p, f(2)= q, f(3)= r; g: B-->A defined by g(p)= 3, g(2)= 2 g(r)= 1. Find gof and fog and show that gof ≠ fog.

19) Let f : X--->Y and g: Y-->Z be two Injective mappings. Then show that their composite gof is injective.

20) If f: A---> B be a mapping and X, Y be sub-sets of A then show that f(X∩ Y) ⊂ f(X) ∩ f(Y).

21) Let f: A--->B be an onto mapping and S, T are subsets of B, show that f⁻¹(S U T) = f⁻¹(S) U f⁻¹(T).

RAW-A

1) When is the mapping f: A--->B said to be one-to-one and onto? Give an example of a mapping f which is one-to-one but not onto.

2) Define the composition of two mappings.

3) Define bijection of a mapping.

 Give an example of a dijective mapping with reasons.

4) Give an example of a mapping f: Z-->Z (Z= set of all integers) which is injective but not subjective.

5) i) if A={5,6,8}, B={2,0,1,4} and f(5)=0, f(6)= 1, f(8)= 4 find whether it defines a mapping.

ii) If A={x, y, a, b}, B={3, 4, 7,8} and f(x)=3, f(y)= 8, f(a)= 9, f(b)= 7. Find whether it defines a mapping.

6) f(x)= x/2 is not a mapping from Z to Z (where Z is set of all integers)- Discuss.

7) The function is defined such that f: R--->R where 

f(x)= 1 if x is rational 

      = -1 if x is irrational.

Find f(2) and f(π).

8) Let the function f: R--->R be defined by f(x)= 2x + 3, x ∈R (R is set of all reals). 

If A={x: 1 ≤ x ≤ 2}, find f(A)

9) a) Are the two mappings f and g defined by

f(x)= (x²-1)/(x -1) and g(x)= x +1 equal? Give reasons.

b) Are the two mappings f: R--->R and g: R--->R defined by f(x)= |x|, g(x)= √x² equal. Give reasons.

10) a) The mapping f: {-2,-1,0,1,2} --> R defined as f(x)= x²+1.

Find f: {-2,-1,0,1,2}. Is it onto?

b) The mapping f:{-1,0,2,5,6} --> {-2,-1,0,11,108} defined as f(x)= x²- x -2. Find whether f is onto p.

c) Show whether the mapping f: Z -->Z,  defined by f(x)= |x|, x ∈Z and Z is the set of all integers is onto or not ? justify your answer.

12) let f: R--->R be defined as f(x)= 2x +3. Show that f is a bijective mapping .

13) Let N be the set of natural numbers . Show that the mapping f: N-->N, defined by f(x)= 2x + 4 for all x ∈ N is an one to one mapping but not onto mapping.

14) Show that the function f: R--->R given by f(x)= |x|, x ∈R is neither nor surjective.

15) Prove that the mapping f: R--->R defined by f(x)= 5x -7 for all x in R is bijective.

16) Examine whether the mapping f: R⁺-->  R⁺ defined by f(x)= x²+1, x∈  R⁺ is 

a) injective

b) surjective.

17) Show that the mapping f: R--->R defined by f(x)=2x +3, x∈ R is bijective.

18) Show that the mapping f: N-->N (N being the set of all natural numbers) defined by f(n)= n +2, n∈ N is one-one but not onto.

19) Show that the mapping f: Q-->Q  defined by f(x)=5x + 2, x∈ Q is bijective where Q is set of all rational numbers.

20) let f be a mapping from A to R defined as f(x)=(1/2) {x + |x|} where A is the set {-5-, -4,-3,-2,0,1,3}. Give reasons whether f is injective mapping or not.

21) Let f: R--->R  where f(x)=x², x∈ R and R = set of real numbers , then show that f is an into mapping.

22) A mapping f: R--->R is defined by f(x)=x² + x, x∈ R. Show that f is neither injective nor surjective.

23) A mapping f: R--->R is defined as f(x)= x³ - x, show that f is onto but not one-one.

24) let f: R--->R be defined by f(x)=3x + 4, x∈ R (R is the set of all real numbers ). Show that f is one-one and onto.

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∩ ⊂ ∉ Φ⊂ ∉ Φ ⊂ ∉ Φ

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COMPOSITION OR OPERATION IN A SET

Let A be any non-empty set. Any mapping from AxA to any set is called composition or operation in the set.

Binary Operation in a set.

Let A be any non-empty set. Any mapping from AxA to A is called a binary Operation or binary Composition in the set A. Thus * be a binary Operation in the set A if and only if a*b ∈ A for all a,b in A.

Identity Element:

Let S be a non-empty set in which a binary operation* is defined.

i) An element say e in S is called left identity element, if e * a = a for all a in S.

ii) An element say e in S is called right identity element, if a * e= a for all a in S.

iii) An element say e in S is called identity element, if it is both left and right identity. 

Inverse Element:

Let S be a non-empty set in which a binary operation* is defined where e is an identity element.

i) If for an element a in S is an element say a' in A such that a' * a= e called left inverse of a. 

ii) If for an element a in S there is an element say a' in S such that a * A' = e, than A' is called right inverse of a.

iii) If for an element say a in S there is an element say a' in S such that a' * a= a * a' = e, then a' is called inverse element.

EXERCISE - A

1) Examine whether the arithmetical addition is a binary relation on the set A={1,0,-1}.

2) Find whether the operation* defined as a * b = a+ b - 2 for all a,b in Z, is a binary operation in Z.

3) In the set Q of all rational numbers an operation * is defined as a * b = a+ b + ab, a, b ∈ Q. Find whether * is a composition in Q. Find whether * is a binary operation in Q.

4) Examine whether usual multiplication is a binary relation in the set A={1, i,-1, - i}.(Where i= √-1). If so find the identity element and inverse of i.

5) (R.") is an algebraic structure. * defined by a * b = 2a + b.

Is it a right identity also?

6) Show that in the set Z={0,1,2,3....) the composition o defined by a o b = |a - b| is a binary relation. Find the identity element and inverse of 4 w.r.t this composition.

7) If the binary operation o be defined on I, the set of all integers by a o b = a+ b +1, a, b ∈ I

i) find the identity element with respect to o.

ii) find the inverse of a with respect to o.

8) Show that in the set

S= {(x     y) x,y,z are integers}}

        o     z 

under the usual matrix multiplication is a binary relation. Find the identity element with respect to this operation.

9) In the set of all 2x2 real matrices of the form 

(x      y 

 -y     x) Show that matrix addition is a binary operation. What is the identity element in this set. Find the inverse of the element 

(2       5

-5       2) in this set.

10) In the set of all positive rational number Q⁺ the composition o is defined as a o b = ab/2, a,b ∈ Q⁺. Show that o is a binary operation in Q⁺.

a) Find , if possible, the Identity element in Q⁺  w.r.t this operation.

b) Find also the inverse of a in Q⁺.

11) Show that ∩ is a binary operation in the power set of S={1,2,3,4}. Find the identity element in the set.

Find the inverse of the {1,2,3} in this power set.

12) Prove that usual multiplication is an operation in the set of all 8th root of unity. What is the identity element in this set.

RAW - A

1) Does addition define a binary operation on the set {1,2,3,4} ?

2) Let the composition * be defined in set of all integers Z by a* b = a - b.  Prove that this composition is binary operation. Find the right identity element in Z with respect to the operation. Is it left identity element ?

3) Let the composition o defined as x o y = x+ y -1 be defined over the set D of all integers. Show that this composition is binary operation in D. Find the identity element in D.

4) Show that in the set of all complex number Z with |Z|= 1 usual multiplication is a binary relation.

5) Prove that in the set of all matrices of the form 

0     a

0     b where a, b are real numbers, usually addition and usual matrix multiplication are binary operations.

6) Prove that is + a binary operation in the set of all continuous functions.

7) Examiner whether the composition * defined in the set of all positive real numbers as a * b= a + log꜀b is a binary relation.

8) Prove that in the set {1, ω, ω²} usual multiplication is a binary operation. Find the identity element with respect to this operation. Find the inverse of ω. 

9) Prove that in the set of real numbers the composition * defined by a * b = aᵇ is not binary relation.       {Hint: -1 * (1/2)= (-1)¹⁾²= √-1 is not real}

10) Prove that in set of all integers the composition * defined by a * b = aᵇ is a binary relation. 

11) Verify whether usual edition is a binary relation in the set {2,4,6,8,10...}.  Find , if possible, the Identity element in this set.

12) Verify whether the usual multiplication is a binary relation in the set {1,3,5,7}. Find, if possible, the Identity element in this set. Find if possible, the inverse element of 5 in this set.

13) Find whether the composition * defined by a * b = a²+ b² is a binary operation in the set of all real numbers.

14) Find whether the composition o defined by a o b = √(a - b) is a binary operation in the set of all rational numbers . Show that there exists no identity element in this set.

15) Show that usual multiplication is a binary operation in the set {1,-1} but not in the {3,-3}.

16) Examine whether the division (÷) is a binary operation in the set {-1,1}. Find , if possible, the identity element with respect to this oparation.

17) Show that usual multiplication is a binary operation in the set A={2,3,4,5...}.  Find, if any, the Identity element in A.

18) Show that in the set of all non-zero real numbers arithmetical multiplication is a binary operation. Does the inverse element exist for all elements in this set.

19) Show that arithmetical addition is a binary operation in Z⁺ (the set of all non negative integers). Find the identity elements (if any). Find the inverse of -6.

20) Show that arithmetical multiplication is a binary in Z⁺ (the set of all non-negative integers). Find the identity element (if any). Find the inverse of 6 in this set.

21) Show that 'addition' is a binary operation in the set of all integers, which are multiple of 5. Find the identity element in this set. What is the inverse of -20 in this set.

22) Find the inverse element of each element in the set A={-1,0,1} with respect to the operation addition.

23) Show that there exists no inverse element of any element except 0 in the Z={0, 1, 2, 3, 5,....} with respect to the binary 'addition'.

24) Show that arithmetic addition is not a binary operation in the set A={-2, 1, 0, 2}.

25) Show that U is a binary operation in the power set of A={1,2,3,4}. Find the identity element in this set. Find the inverse of the element {2,3,4} in this set.

26) Show that the composition o define as a o b = a+ b +1 is a binary operation in the set of all odd integers but it is not a binary operation in the set of all even integers. Find the identity element in the form set.

27) Show that the 'matrix multiplication' is a binary operation in the set

M={(a     b) 

        c     d) : a,b,c,d are integers}. Find the identity element in M.

28) Prove that usual multiplication is a binary operation in the set of all non zero complex numbers. Find , if any, the Identity element in this set.

 Find , if any, the inverse elements of 1- i.

29) Prove that usual matrix multiplication is a binary operation in the set 

{(a     b

   a    b) : a,b are real and a+ b ≠0}

Find , if possible, the Identity element in this set.

30) Prove that usual multiplication is a binary operation in the of all of 5th root of unity. Find , if any, the Identity element this set.

1) Yes 

2) 0 is right identity; This is not left identity.

3) 1 7)  no  8) w² 

9) Hint: -1 * (1/2)= √(-1) =√1 is not real

11) yes, not possible

12)  yes, it is binary relation; 1 ;  not possible 

13) yes 

16) binary operation; 1 is the identity element

17) no, identity element exist 

18) Every element has an inverse 

19) identity element is 0; 0 is the inverse of -6

20) identity element 1, no inverse

21) 0 is identity element; 20 is inverse of  - 20 

22) inverse of -1, 0, 1 are respectively 1, 0, - 1 

23) ∅ is identity element; No inverse 

26) -1 

27) 1     0

       0     1 is identity element 

28) 1 is identity element; (1/2) (1+ i) is inverse of 1 - i.

29) there is no identity element in the set.

30) 1

GROUP THEORY 

Binary composition (operation).

If G be a non empty set and a, b ∈G then a composition denoted as o such that a o b ∈ G is called binary composite in the set G.

 In other words a binary composition in a set G is a mapping of G x G into G which associates to each ordered pair (a,b) of member of G, a member of G.

Various Types of Compositions 

a) Associative composition, 

In a set G the composition denoted as o is said to be associated if,

(a o b) o c = a o (b o c) ∀ a, b, c ∈ G

In this case algebraic structure (G, o) where G is the set and o  the composition is said to be associative.

b) Composition with identity element:

 In a set G the composition denoted as o is said to be composition with identity element denoted by e if there exists an element e belonging to G such that 

a o e = e o a ∀ a ∈ G.

The element e ∈ G is called The Identity element and the algebraic structure (G, o) is said to be having identity element.

c) Existence of Inverse 

Invertible elements in a equipped with a composition and having an identity element 

It should be noted that inverse of an element must belong to the set.

d) Commutative composition 

In the set G a composition denoted by o is said to be commutative if for all a, b ∈ G

a o b = b o a

and in this case the algebraic structure (G, o) is said to be committed or abellion.

Groups 

A non empty set G with a binary composition denoted by o and satisfying the following properties is called a group.

G₁ : The composition is associative .

i.e., (a o b) o c = a o (b o c) ∀ a, b, c  ∈ G.

G₂: Existence of Identity .

∀  ∈ G there exist and unique  (to be proved later on) element e ∈ G such that 

a o e = e o a = a

G₃: Existence of inverse.

∀ a ∈ G there exists an element b ∈ G such that 

a o b = b o a = e (the identity element).

The element b is called inverse of a and is denoted by a⁻¹ so that we can say that 

a⁻¹ o a = a o a⁻¹ = e

Abelian Group 

If an addition to the above three properties the binary composition in G is commutative.

i.e., a o b = b o a ∀ a, b ∈ G then the group G is called abelian or commutative.

Note :

In case there exists at least one pair of elements a, b ∈ G for which a o b ≠ b o a then the group (G, o)  is said to be non-abellion or non-commutative group.

Some Defination.

a) A set G with a binary composition is said to be a groupoid.

b) A set G with a binary composition which is associative is said to be semi-group 

c) A set G with a binary composition which is associative and identity element exists is said to be monoid.

EXERCISE - A

1) The set I of all integers, i.e., {........-4,-3,-3,-1,0,1,2,....} is a group with respect to the operation of addition.

2) The set I integers, i.e., {-4,-3,-3,-1,0,1,2} is not a group with respect to the operation of addition.

3) G={-2,-1,1,2} does not form a group for multiplication.

4) E={The set of all even integers including zero}, for + is a abelian Group 

5) O={The set of all odd integers}, for + is not a group 

6) I={The set of all integers} for multiplication is a group or not.

7) I={The set of all integers), for - (substraction) is a group or not.

8) Q={The set of all rational numbers including zero} for

a) multiplication 

b) addition 

c) G={x: 0< x≤1} for multiplication.

d) division 

Is group or not 

9) Q₀= {The set of non zero rational numbers}.

a) for multiplication 

b) for addition.

c) for division 

Is group or not 

9) G={The set of multiple of all integers by a fixed integer m}, for +

Is group or not 

10)a) Prove that the set

G={.....2⁻⁴, 2⁻³, 2⁻², 2⁻¹, 1, 2, 2², 2³, 2⁴....}

forms and infinite abelian group w.r.t., multiplication.

b) Show that the set of numbers of the form a+ b √3 where a and b are rationals is a group for the composition of addition.

11)a) Prove that the set G₀ of all non-zero complex numbers is a group w.r.t. multiplication of complex numbers.

C₀ {Z: Z = a+ ib where a, b ∈ R and both a and b together are not zero}

12) Show that the set of all complex numbers z such that |z|= 1 forms a group with respect to multiplication of complex numbers.

13) Show that the following sets form a group under ordinary multiplication.

a) G= {a + b √2: a, b ∈ Q}

b) G={a + ib : a, b ∈ Q}

Where a and b together are not zero.

14) Prove that the sets of Question(13) constitute a group under addition as the composition.

15) The set A={1,ω, ω²}, where 1, ω, ω² are cube roots of unity and ω³= 1 forms a group w.r.t. multiplication composition.

16) Prove that the set consisting of the four fourth roots of unity forms and abelian group w.r.t. multiplication composition.

17) Prove that the set of all vectors in space do not form a group with respect to vector multiplication as composition, but is a group w.r.t. vector addition.

18) Prove that the set of all m x n matrices having their elements as integers (rational, real or complex numbers is an infinite abelian group with matrix addition as composition.

12) Prove that the set of all n x n non singular matrices having their elements as rational (real or complex) numbers is an infinite non abelian group with matrix multiplication as composition. If the elements were integers then will it be a group?

13) Show that the set of matrices 

A= cos a      - sin a

      sin a         cos a

Where a is real (integral or rational) form a commutative group relative to matrix multiplication.

14) If G= a    0    or     1     a

                0    0             0     1 are the matrices and a is any non zero real number 

Show that G is a commutative group under multiplication.

15) Prove that in a matrix group under multiplication either all the matrices are non singular or all are singular.

Show that the set of all matrices of the form 

x      x

x  where x is a non zero real number is a group of singular matrices for multiplication. Find the identity and inverse of an element.

16) Prove that the set G={ cosθ + i sinθ) : θ runs over all rational numbers} forms and infinite abelian group with respect to ordinary multiplication.

17) Show that the set Q of all rational numbers other than 1 with the operation defined by a o b = a+ b - ab constitutes an abelian group.

18) Q₋₁={set of all rational numbers other than -1} is an abelian group for the composition defined as 

a o b = a+ b + ab

Identity is zero, a⁻¹ = -a/(a+ 1) (a≠ -1)

Note: In case the set be of integers in place of Q₋₁ then it will not form a group for the same composition as a⁻¹ = -a/(a+1) is not an integer.

19) I={set of all integers for the composition as a o b = a + b +1 is also an abelian group.

20) Show that the set N of natural numbers for the binary composition a o b = a+ b + ab is a semi group.

21) Prove that the set Q⁺ of all positive rational numbers forms an abelian group for the composition o defined as 

a) a o b = ab/2

b) (a o b)= ab/7

c) (a o b)= ab/4

22) Let G be a set of elements on which an algebraic operation x is defined such that a x b ∈ G ∀ a, b ∈ G. Show that G is an abelian group for the operation if the following postulates are satisfied.

a) (a x b) x c = a x (c x b) ∀ a, b, c ∈ G.

b) There exists a left identity e ∈ G such that 

e x a = a ∀ ∈G.

c) Corresponding to every element a ∈ G there exists a left inverse a⁻¹ ∈G such that a⁻¹ x a = e.

23) S is a nonvoid set and A and B are sub-sets of S which belongs to the power set P(S). An operation o is defined on the elements of P(S) as

A o B = A U B

Verify whether P(S) will be a group or not.

24) Show that the set of all n, n-th roots of unity forms a finite abelian group G under multiplication as composition.

25) The adjoining table defines a certain binary operation o on the set A={ab,c,d,e}. Show that the set A forms a group w.r.t. the binary operation o and find the inverse of each element.

26) Show that the four matrices

A= 1  0 B= -1  0 C= 1   0 D= -1   0

      0  1       0   1      0  -1        0  -1

form a Multiplicative group. Is it abelian or not ?

27) Show that the set of four transformations f₁, f₂, f₃, f₄, on the set of complex numbers defined by 

 f₁(z)= z, f₂(z)= -z, f₃(z)= 1/z, f₄(z)= -1/z

Forms a finite abelian group w.r.t. the composition known as composition of two functions of product of two functions.

28) Show that the set of six transformation  f₁, f₂, .....f₆ on the set of complex numbers defined by 

f₁(x)= x, f₂(x)= 1/x, f₃(x)= 1- x, f₄(x)= x/(1- x), f₅(x)= 1/(1- x), f(x)₆= (x -1)/x.

Forms a finite non abelian Group of order 6 w.r.t composite operation.

29) Verify whether the set G for the defined composition o is group or not.

G= {aᵢ : i ∈ N and  0 ≤ i < 7}

and aᵢ o aⱼ = aᵢ₊ⱼ {where i+ j < 7}

 And aᵢ o aⱼ = aᵢ₊ⱼ₋₇ {where i + j ≥ 7}

30) Show that the set G==0.1,2,3,4,5} is a finite abelian group under ordinary addition reduced modulo 6 as the composition.

31) Is the set {1,2,3,4,5} a group under addition modulo 6.

32) Show that the set {1,2,3,4} of four integers is a finite abelian group under multiplication modulo 5.

33) Is the set {1,2,3,4,5} of five integers a group under the composition of multiplication modulo 6.

34) Show that the following sets are groups.

a) {0,3,6,9} for addition modulo 12.

b) {1,5,7,11} for multiplication modulo 12.

35) Show that the residue classes {1}, {3}, {5}, {7} modulo 8 forms a Multiplicative group.

General Properties of Groups 

Let G be a group and o be the binary o be the binary operation; then a o b will be written as ab using multiplication notation. The identity will be denoted by e and the universe of a will be denoted by a⁻¹. However if we use additive + for a then e will be replaced by 0(zero) and a⁻¹ by - a.

Property 1. Left cancellation law.

If a, b, c ∈G then ab= ac => b = c

Property 2: The left identity element e is also the right identity element.

Property 3: The left inverse of an element is also its right inverse.

Property 4: Right Cancellation Law 

Property 5: The equation ax= b and ya= b have unique solution in G where a,b ∈ G.

Property 6: The identity element e in a group is unique.

Property 7: The inverse of an element in a group is unique 

Property 8: The inverse of a is a⁻¹. 

Property 9: Reversal Law

EXERCISE - B

1) If every element of a group of a group is its own inverse then show that G is abelian 

2) If b⁻¹a⁻¹ ba = e ∀a, b ∈ G then the group G is abelian.

3) show that the composition table for a finite group contains each group element once and only once in each of its rows and columns.

Property 10: An alternative definition of a group.

A set G a composition is a group iff

i) The composition is associative.

ii) The equation ax= b and ay= b, a, b ∈ G have unique solution in G.

∩ ⊂ ∉ Φ⊂ ∉ Φ ⊂ ∉ Φ

∪⊄#𝔓∪∩.  ⊆ ⊇decimal Atul without.   ∈ ∪ ∩  ⊂ ∉ Φ ₁ ₂ ⁺ f⁻¹(y ∀

RING & FIELD 

Let R be a non-empty set; + and • be two composition in it. R is said to be a Ring with respect to these two compositions, if the following axioms hold:

I. R is an abelian group with respect to the first composition+ that is 

i) for all elements a,b in R, a+ b ∈R [closure property] i.e + is a binary operation in R.

ii) for all elements a,b and c in Ra+ (b + c)= (a+ b)+ c,    (Associative property)

iii) there exists an element 0, called zero elements, in R such that 0 + a = a for all a in R.

iv) corresponding to an element a in R, there exists an element -a in R such that -a + a = 0. (-a is called additive inverse of a).

v) for all elements a,b in R, a+ b = b + a (commutative property)

II

i) for all elements a,b in R, a,b ∈ R

ii) for all elements a,b and c in R, a.(b.c)= (a.b).c

III 

i) for all elements a, b, c in R a,(b+ c)= a.b + a.c  (left distributive property)

ii) For all elera,b, c in R. (b+ c).a= b.a + c.a (Right distributive property)

Note:

1) In notation we write "(R, +, •) is a ring", if R is a ring with respect to+ and •

2) A ring R may contain finite or infinite number of elements 

3) The composition - and • are called 'addition' and 'multiplication' respectively. But these are not necessarily arithmetical addition and multiplication.

Commutative Ring

A ring R with respect to the composition+ and • is said to be commutative Ring if a,b = b.a for a, b in R.

Ring with unity 

A ring R with respect to the composition + and • is said to be 'Ring with unity ', if there exists an element e in R such that e.a = a.e = a for all a in R. The element e is called the unity elements in R.

Elementary properties of Ring.

Since Ring is a Group with respect to +, all the properties of group hold. For example, a+ b = a+ c implies b = c; the zero element 0 is unique etc. Besides those we get more properties:

PROPERTY 1

In a Ring (R, +, •)

a.0= 0 and 0.a= 0 for all a in R. Where 0 is zero element in the ring.

PROPERTY 2:

In a ring (R, +, •)

i) a.b= -(a.b)

ii) (-a).b = -(a.b)

iii) (-a).-(b)= a.b

PROPERTY 3:

In a ring (R, +, •)

a- b is defined as a - b = a+(-b). Then

i) (a - b).c= a.c - b.c

ii) c.(a - b)= c.a - c.b

PROPERTY 4:

In a ring with unity (R, +, •), the unity element e is unique.

SUB-RING

Let R be a ring with respect to the composition + and • ; H be a non-empty subset of R. If H itself be a Ring with respect to the same composition + and •, then H is called a subring of (R, +, •).

FIELD 

Let F be a non-empty set, + and • be two composition in it. F is said to be Field with respect to these composition if following axioms hold:

I. Field is an abelian group with respect to the first composition +, that is 

i) for all elements a,b in F, a+ b ∈ F i.e., + is a binary operation in F.

ii) for all elements a,b, c in F, a+ (b +c)= (a+ b)+ c

iii) there exists an element 0 is called zero element, in F such that 0+ a = a for all a in F.

iv) corresponding to an element a in F there exists an element - a in F such that -a + a= 0.

v) for all elements a,b in F. a+ b = b + a

II 

i) for all elements a, b in F, a.b ∈ F

ii) for all elements a,b and c in F, a.(b.c)= (a.b).c

iii) there exists an element e, called identity element (or multiplicative identity) in F such that e.a= a for all a in F

iv) corresponding to an element a ≠ 0 in F, there exists an element A' in F such that a'.e = e and A' is called multiplicative inverse of a.

v) for all elements a,b in F a.b= b.a

III 

For all elements a,b, c in F we have a.(b + c)= a.b + a.c

Note

1) In notation we write "(F, +, •) is a Field", if F is a Field w.r.t the composition + and •.

2) The composition + and • are called 'addition' and 'multiplication' respectively but these are not necessarily arithmetic addition and multiplication.

3) From the axioms stated in II above we can say the set F - {0} is an abelian group with respect to the second composition.

4) The axioms I, (i), (ii), (v) of II and III show that a Field is a Ring. Moreover axioms I, (i), (ii), (iii), and (v) of II and III show that a Field is a Commutative Ring with unity element e.

EXERCISE - A

1) Show that the set of all real numbers is a ring with respect to arithmatical addition and arithmetical multiplication.

2) Show that the set of all complex numbers is a ring with respect to usual addition(+) and usual multiplication (*).

3) Show that the set {z: z is a complex number and |z|= 1} is not a subring of the ring of all complex numbers under arithmetical addition and multiplication.

4) Show that the set

M{[x    y

     z    u]} : x, y, z, u ∈ Z denotes set of integers, form a ring with unity.

5) i) Show that the set of all 2x2 real matrices is a ring under matrix addition and matrix multiplication. Is it commutative Ring ? Is it ring with unity ?

ii) Show that the set of all 2x2 real matrices of the form 

(a       b

0        c) is a subring of the ring of all 2x2 real matrices under matrix addition and multiplication.

6) Let ⊕ and ⊚ be defined on the set of all integers (z) by the rule a ⊕ b = a+ b -1 and a ⊚ b = a+ b - ab for a, b ∈z. Prove that (z, ⊕, ⊚) is a ring with unity. Is it a Commutative Ring?

7) Prove that the set of all matrices 

M= {(a    b 

        2b   a)  : a, b are rationals} is a ring with respect to matrix addition and matrix multiplication. Is it is a ring with unity ?

8) Given that the set of all 2x2 matrices of the form 

x     y

0     z where x, y, z are integers form a ring with respect to matrix addition and multiplication. Is it a Commutative Ring? Justify your answer.

9) Show that the set {a + b √2, where a, b are integers} forms a commutative Ring with the identity element, with respect to addition and multiplication. Is it Commutative Ring with unity element?

10) Prove that the commutative group of all real, numbers of the form a+ b √2 (a, b are rational numbers) under usual addition is a Field under usual multiplication.

11) Show that the set of real numbers is a field under numerical addition and numerical multiplication.

12) Show that the set of all complex numbers is a Field under numerical addition and multiplication.

13) Show that the set of all 2x2 real matrices of the form 

x      y

-y     x forms a Field with respect to matrix addition and multiplication.

14) Considering the set of all matrices 

M= {(a     b

         2b   a) : a,b are rational} is a Ring under matrix addition and matrix multiplication, show that M form a Field.

15) Show that the set 

s={(0     0

       x     y) : x,y are integers} is a Ring but not a Field under usual matrix addition and multiplication.

16) Show that the Ring of matrix 

[(a        b

  2b      a): a, b ∈R] doesn't form a Field. (R = set of all real numbers).

17) Let S=[ a+ b √5: a, b are rational number]. Show that S is a subfield of the field (R, +, •) of real numbers.

18) Prove that the set S={a+ b ω: a,b are real numbers} is a subfield of the field of all complex numbers under usual addition and multiplication.

19) Prove that in a ring (R, +, •) with the unity element 1, (-1).a = - a where a ∈ R.

20) If a,b are element in Ring (R, +, •), examine if the equality (a + b). (a - b)= a²- b¹ holds where a²= a.a

21) In a ring prove -(a + b)= - a - b where a,b ∈ the ring.

22) Show that in a Field F.

i) (-x)⁻¹ = -x⁻¹.

ii) (-a)(-b)⁻¹= ab⁻¹ where x⁻¹ is Multiplicative inverse of x≠ 0.

VECTOR SPACE 

∩ ⊂ ∉ Φ⊂ ∉ Φ ⊂ ∉ Φ

∪⊄#𝔓∪∩. ⊆ ⊇decimal Atul without. ∈ ∪ ∩ ⊂ ∉ Φ ₁ ₂ ⁺ f⁻¹(y ∀ ⊕ ⊚