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...