Complement Of a Set –
If A is a subset of a universal ξ then the set of all those elements of ξ which do not belong to A is the complement of A and is noted by A’ or A̅ or Aᶜ.
Thus, A’ = ξ – A = {x ǀ x ∈ ξ, x ∉ A}
We may also simply say A’ = {x ǀ x ∉ A}. since it is understood that x ∈ ξ
For Example –
(i) Let, ξ be the set of all pupils of a class and A the set of all girls in that class, then A’ is the of all boys in that class.
(ii) If, ξ = the set of counting numbers and A = set of counting numbers less than 100, then A’ = {100, 101, 102,……….}
(iii) If, ξ = set of natural numbers, A = set of even natural numbers, then A’ = set of odd numbers.
(iv) Let, ξ = the set of English alphabet and let V = {x ǀ x is a vowel of the English alphabets}, then V’ = {x ǀ x is a consonant of the English alphabet}
(v) (a) Let, ξ = {1, 2, 3, 4, 5, 6, 7, 8}, and A = {1, 3, 5, 7}, A’ = {2, 4, 6, 8}
Also, n(ξ) = 8, n(A) = 4,
Then, n(A’) = n(ξ) – n(A) = 8 – 4 = 4
(b) If, n(ξ) = 40, n(A) = 15, n(B) = 20,
then n(A’) = n(ξ) – n(A) = 40 – 15 = 25
also, n(B’) = n(ξ) – n(B) = 40 – 20 = 20
it is obvious that the complements of A’ is A, that is, (A’)’ = A. The complement of the universal set is the empty set, that is ξ’ = ϕ and the complement of the empty set is the universal set, i.e., ϕ’ = ξ
Laws Of The Complement Of The Set.
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