Last Time Revision Set Theory (2)

1) Express each of the following statements in set-theoretic notation:

a) x is belongs to the set A.              x ∈ A

b) Set A is a subset of set B.        A⊆B

c) P is a proper subset of Q.       P⊂Q

d) a is not an element of set X.       a ∉ X

e) Y is not a proper subset of Z.       Y ⊄ Z

f) A set whose elements are contained in U but not contain in B.    B' 

2) Suppose P and Q are two given sets and the corresponding Universal set U; express each of the following statements in symbols:

a) A set whose elements are contained either in P or in Q

b) A set which contains the common elements of P and Q.

c) A set whose elements belongs to U but do not belongs to Q.

d) A set whose elements belongs to Q but do not belongs to P.

e) A set whose elements do not belong to any of the sets P and Q.

f) A set which does not contain the common elements of P and Q .

3) State the meaning of each of the following set notations :

a) A ⊆B

b) p ∉ P

c) A⊃ B

d) A ⊄ B

e) X={0,∅ } 

f) 2 ∈ Z

4) Represent the following sets in roster form:

a) set of vowels in English alphabet.

b) set a factors of the number 36.

c) A={x:x is a prime integers and 6< x ≤ 29}

d) P={x:x ∈ N and x ≤ 12}.

e)  set of the letters in the word 'statistics'.

5) Rewrite the following sets in the set-builder form:

a) A={....,-3,-2,-1,0,1,2,3....}.

b) P={1,3,5,7,9,11}

c= set of roots of the equation x⁴-13x²+36=0.

d) Set of even positive integers greater than 4 and less than or equal to 19.

6)  State whether each of the following sets is finit or infinite:

a) D={x: x is the number of people living on the earth}.

b) W={x:x is the time a person waits for a bus}.

c) V={ x:x is an odd integers exceeding 889}.

7) State whether the following sets are equal:

a) X={∅} ; Y=∅

b) A={-2, -5}; B={x:x is a root of the equation x²+7x+10=0}

c) P={b, i, e,e,n,s,u} ; Q={x:x is a letter in the word 'Business'}.

d) A={x:x is a digit in the number 30255} ; B={x:x is an integers and 0≤x≤5} ; c={1,0,2, 3, 4,5}.

8) Some well defineds sets are given below; identify the null sets:

a) A={0}

b) B={∅}

c) C= ∅

d) set of boy students in a girls' college.

e) P={x:x is an integers 1< x ≤2}

f) R={x:x is a positive and x²+7x+12=0}.

g) A={x:x is an integer and 6x²-5x +1=0}

9) State which of the following are null sets :

a) {x:x 3x²-4=0, x is an integer}.

b) {x: (x+3)(x+3)=9, x is a real number}

c) A ∩B - A.

10) State which of the following statements are correct or incorrect:

a)  If P∪Q={a,b,c,d}, then a∈ P ∧ a ∈Q.

b) If P∩ Q={a,b,c,d}, then a∈ P ∧ a ∈Q.    

c) If A ∩B= ∅, then A and B are disjoint sets.

d) If A ⊆ B, then B⊆ A.

e) If A ⊆ B and B ⊆ C, then A ⊆ C

f) x ∈ A ∪ B => x ∈ A.

g) If A ⊆ B and B ⊆ A, then A= B.

h) If A={2, 4, 6, 8}, then {2,4} ∈ A.

i) If A={2,4,6,8}, then {2,6,8} ⊆A

j) If A={2,4,6,8}, then ∅ ⊆ A

k) If A={2,4,6,8}, then {2,4} ⊂ P(A).

l) (A - B), A ∩B and (B - A) are mutually disjoint

m) If A= {2,4,6,8}, then {2,6,8} ∈ P(A).

11) If a ∈ A and a∈ B, does it follow A  ⊆ B?

12) State with reasons , which of the following statements are true or false:

a) {a} ∈ {a,b,c}

b) a  ∈ {a,b,c}

c) a ⊂ {a,b,c}.

d) a ∉ {a,b,c}

13) Which of the following statements are correct/incorrect ?

a) 3 ⊂ {1, 3, 5}

b) 3 ∈ {1, 3, 5} 

c) {3} ⊂ {1, 3, 5}

d) {3} ∈ {1, 3, 5}

14) If A={a,b,c} name

a) the subsets

b) The proper subsets of A.

15) Define power set of a set A. Find the power set of A={a,b,c}. If B be the power set of A, state with reason, which of the following statement is correct :

A ⊃ B, A ∈ B, A ⊂ B, A= B, A⊄ B.

16) Fill in the gaps:

a) The number of subsets in a set consisting of four distinct elements are ___.

b) The number of proper subsets in a set consisting of n distinct elements are ___.

c) If x∈ A => x ∈ B then _____.

d) If A ⊆B and B⊆ A then _____.

e) The set P whose elements are all subsets of the set {1,2} is given by

P={___, ___, ____, ___}

f) If A and B are disjoint sets then n(A ∪B)= ___.

g) If A∪ B= A ∩B then _____.

h) The dual of A ∪(B ∩C) = (A∪B) ∩(A ∪C) is ____.

i) The dual of A ∪ U= U is _____.

j) If A is a given set and ∅ is the null set then ____.

17) Let A={a,b,c}, B={a,b}, C={a,b,d}, D={c,d} and E={d}. State which of the following statements are correct:

a) B ⊂ A 

b) D ⊅ E

c) D ⊂ B

d) {a}⊂ A

18) Let A={a,b,c,d,e, f ,g ,h, i}, B={b,d,f, h}, C={a, c, e, g, i}, D={c, d, e} and E={c, e}.

Which set can equal X if we are given the following informations ?

a) X and B are disjoint

b) X⊂ A but X ⊄ C

c) X  ⊂ D but X ⊄ B

d) X ⊂ C but X ⊄ A.