element of C. Example Let A = { a, e, i, o, u } and B = { a, b, c, d }. Is A a subset of B ? No.
(Why?). Is B a subset of A? No. (Why?) Example Let A, B and C be three sets.
If A ∈ B and B ⊂ C, is it true that A ⊂ C?. If not, give an example. Solution No. Let A = { }, B = {{ }, } and C = {{ }, , }.
Here A ∈ B as A = { } and B ⊂ C. But A ⊄ C as ∈ A and ∉ C. Note that an element of a set can never be a subset of itself. .
. Subsets of set of real numbers As noted in Section . , there are many important subsets of R . We give below the names of some of these subsets.
The set of natural numbers N = { , , , , , . . .} The set of integers Z = {. .
., – , – , – , , , , , . . .} The set of rational numbers Q = { x : x = p q , p, q ∈ Z and q ≠ } SETS which is read “ Q is the set of all numbers x such that x equals the quotient p q , where p and q are integers and q is not zero”. Members of Q include – (which can be expressed as – ) , , (which can be expressed as ) and – The set of irrational numbers, denoted by T , is composed of all other real numbers.
Thus T = { x : x ∈ R and x ∉ Q }, i.e., all real numbers that are not rational. Members of T include , and π . Some of the obvious relations among these subsets are: N ⊂ Z ⊂ Q , Q ⊂ R , T ⊂ R , N ⊄ T