Showing Equivalence using Set Properties 1

In this class, We discuss Showing Equivalence using Set Properties 1.

The reader should have prior knowledge of set theory basics. Click Here

Example 1:

A – (B–C) = (A – B) ∪ (A∩C)

Solution:

A -(B–C)

A – (B ∩ C’)

A ∩(B ∩ C’)’

A ∩ (B’ ∪ C) from de Morgan law

(A ∩ B’) ∪ (A∩C) from distributive law

(A – B) ∪ (A ∩ C)

Example 2:

If A ∪ B = A ∪ C and A ∩ B = A ∩ C then show that B = C

Solution:

Take A ∪ B = A ∪ C

B ∩ (A ∪ B) = B ∩(A ∪ C)

Take LHS B ∩ (A ∪ B)

= B since B ⊆ A ∪ B

Take RHS B ∩(A ∪ C)

(B ∩ A) ∪ ( B ∩ C) from distributive law

(A ∩ B) ∪ ( B ∩ C) from commutative law

(A ∩ C) ∪ ( B ∩ C)

(A ∪ B) ∩ C

(A ∪ C) ∩ C

= C since C ⊆ A ∪ C