LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

COMPLEMENT OF A SET

**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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