Question

Use only set identities (i.e., no membership tables or Venn diagrams) to demonstrate
that:
((? ∩ ?) ∩ ?) ∪ (? ∩ ?) ∪ (((? ∪ ?) − (? ∩ ?)) − ?)
and
(((C"complement" − (? ∪ ?)) ∪ ?) − ?) ∪ (? ∩ ? ∩ ?)

Answer 1

((Aꓵ C) ꓵ B) U (CꓵB) U (((A U B) - (A ꓵ B)) - C)

Apply Associative law ( A ꓵ C) ꓵ B= A ꓵ (C ꓵ B)

(Aꓵ (C ꓵ B)) U (CꓵB) U (((A U B) - (A ꓵ B)) - C)

APPLY A- B= A ꓵ B’

(Aꓵ (C ꓵ B)) U (CꓵB) U (((A U B) ꓵ (A ꓵ B)’) ꓵ C’)

(C ꓵ B) U ( (A U B ) ꓵ (A ꓵ B)’ ꓵ C’)

(Remember (X ꓵ Y) U Y =Y)

=( (CꓵB) U (AUB)) ꓵ ((CꓵB) U (AꓵB)’) ꓵ ( ( C ꓵ B) U C’))

Applying distributive law

A U (B ꓵ C ꓵ D)= (A U B) ꓵ (AU C) ꓵ (AUD)

=(C U A U B) ꓵ ( B U A U B) ꓵ (C U (A ꓵ B)’) ꓵ (B U (AꓵB)’) ꓵ (C U C’) ꓵ (B U C’)

=(A U B U C) ꓵ (A U B) ꓵ ((AꓵB)’ U C) ꓵ Y ꓵ Y (B U C’)

WE APPLY TWO LAWS

AUB=BUA communtative law

XU Y=Y

=(AU B U C) ꓵ ( (AꓵB)’ U C) ꓵ ( B U C’)

NOW TAKE THE SECOND ONE

(((C"complement" − (? ∪ ?)) ∪ ?) − ?) ∪ (? ∩ ? ∩ ?)

=(((C’- (AUB)) U C)-B) U (A ꓵ B ꓵ C’))’

=(C’ ꓵ (AUB)’ U C) ꓵ B’)’ ꓵ (AꓵBꓵC’)’

WE USE A-B= A ꓵ B’

((C’ ꓵ (A U B)’ U C)’ U B) ꓵ ((A ꓵB)’ U C)

DEMORGANS LAW

A’U B’=(AꓵB)’

AND

INVOLUTION LAW

(A’)’=A

( ( ( C U A U B) ꓵ C’) U B) ꓵ ((AꓵB)’ U C)

ASSOCIATIVE LAW

((AUBUC) U B) ꓵ (C’ U B) ꓵ ((AꓵB)’ U C)

DISTRIBUTIVE AND COMMUTATIVE LAW

(A U B U C) ꓵ ( B U C’) ꓵ ((AUB)’ U C)

USING COMMUTATIVE LAW

Check the both underlined statement

Given two expressions are equivalent