every set A is a subset of itself, i.e., A ⊂ A. Since the empty set φ has no elements, we agree to say that φ is a subset of every set . We now consider some examples : MATHEMATICS The set Q of rational numbers is a subset of the set R of real numbes, and we write Q ⊂ R. If A is the set of all divisors of and B the set of all prime divisors of , then B is a subset of A and we write B ⊂ A.
(iii) Let A = { , , } and B = { x : x is an odd natural number less than }. Then A ⊂ B and B ⊂ A and hence A = B. (iv) Let A = { a, e, i, o, u } and B = { a, b, c, d }. Then A is not a subset of B, also B is not a subset of A.
Let A and B be two sets. If A ⊂ B and A ≠ B , then A is called a proper subset of B and B is called superset of A. For example, A = { , , } is a proper subset of B = { , , , }. If a set A has only one element, we call it a singleton set .
Thus,{ a } is a singleton set. Example Consider the sets φ , A = { , }, B = { , , }, C = { , , , , }. Insert the symbol ⊂ or ⊄ between each of the following pair of sets: (i) φ . .
. . C Solution φ ⊂ B as φ is a subset of every set. A ⊄ B as ∈ A and ∉ B (iii) A ⊂ C as , ∈ A also belongs to C (iv) B ⊂ C as each element of B is also an