Sets and Relations

Complement of a set:-

The complement of the set A denoted by

A^c oror A’ is formed by removing each element of A from the universal set .Thus is μ is the 

universal set,

A^c = μ - A

A^c = {x | x ∉ A}  

By definition A ∩ A^c = φ

A∩A^c = μ

|A^c|=  |μ|-|A|

|A^c|+ |A|= |μ|

Prove that: A-B = A ∩ B^c

A-B ={x |x ϵ A and x ∉ B}  

= {x |x ϵ A and x ∉ B^c}

= A ∩ B^c

B-A = B ∩ A^c (imp results)

A-B = A ∩ B^c

De Morgan’s Laws:-

For any two sets A, B

(i) (A U B)^c = A^c ∩ B^c

(ii) (A ∩ B)^c = A^c U B^c

(iii) A – (B U C) = (A - B) ∩ (A - C)

(iv) A – (B ∩ C) = (A - B) U (A - C)

(v) |A^c|^c = A

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

(i) A^c

(ii)(A ∩ B)^c = A^c U B^c

(iii) A^c ∩ B^c = (A U B)^c

Note:

|A U B U C|= |A ∩ B^c ∩ C^c |+ | A^c ∩ B ∩ C^c| + A^c ∩ B^c ∩ C) + |A U B U C^c | + |A ∩ B^c ∩ C^c |+ | A^c ∩ 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 A_n|

Principle of inclusion and exclusion:-

|A^C₁ ∩ A^C₂ ∩ --- A^C_n|= |μ|- |A₁ U A₂ U --- U A_n|