# Sets & Relations

Complement of a set: The complement of the set A denoted by Ac or |A̅| or A’ is formed by removing each element of A from the universal set .Thus is μ is the universal set,

Ac = μ – A

Ac = {x | x ∉ A}

By definition A ∩ Ac = φ

A∩Ac = μ

|Ac|= |μ|-|A|

|Ac|+ |A|= |μ|

Prove that: A-B = A ∩ Bc

A-B ={x |x ϵ A and x ∉ B}

= {x |x ϵ A and x ∉ Bc}

= A ∩ Bc

B-A = B ∩ Ac (imp results)

A-B = A ∩ Bc

De Morgan’s Laws: For any two sets A, B

(i) (A U B)c = Ac ∩ Bc

(ii) (A ∩ B)c = Ac U Bc

(iii) A – (B U C) = (A – B) ∩ (A – C)

(iv) A – (B ∩ C) = (A – B) U (A – C)

(v) |Ac|c = A

Given two non-empty sets A, B which are not disjoint then represents the following sets in venn diagram.

(i) Ac(ii)(A ∩ B)c = Ac U Bc(iii) Ac ∩ Bc = (A U B)cNote: |A U B U C|= |A ∩ Bc ∩ Cc |+ | Ac ∩ B ∩ Cc| + Ac ∩ Bc ∩ C) + |A U B U Cc | + |A ∩ Bc ∩ Cc |+ | Ac ∩ B ∩ C| + |A ∩ B ∩ C|

For any 3 set A, B, C

|A U B U C|= |A|+ |B|+|C|- |A∩B|- |B∩C|-|A∩C|+|A∩B∩C|

A, B, C, D are any 4 sets

|A U B U C U D|= | (A U B) U (C U D)|

=|A U B|+ |C U D|- | (A U B) ∩ (C U D)|

= |A|+ |B| – |A∩B|+ |C|+|D|- |C ∩ D|- |A ∩ D| – |B ∩ C|- |A ∩ B ∩ C|+|A ∩ B ∩ D|+ |A ∩ C ∩ D|+|B ∩ C ∩ D|- |B ∩ D|- |A ∩ C|- |A ∩ B ∩ C ∩ D|

Note: |A1 U A2 U A3 — U An|

$$\sum\limits_{n=1}^{i}{\left| Ai \right|-\sum\limits_{i\le j\le k}{\left| Ai\cap Aj \right|+}\sum\limits_{i\le j\le n}{\left| Ai\cap Aj\cap Ak \right|}+{{\left( -1 \right)}^{n-1}}\left| {{A}_{1}}{{A}_{2}}{{A}_{3}}—-{{A}_{n}} \right|}$$.

Principle of inclusion and exclusion:-

|AC1 ∩ AC2 ∩ — ACn|= |μ|- |A1 U A2 U — U An|