Question

Basic Rules (Equalities) in Set Algebra Associative Laws: Commutative Laws: Distributive Laws: De Morgan's Laws: Complement Laws: X n (Y U Z) = (X Y) U (X n Z) Repetition: 0/1 Laws: Identity: Bound Laws:

5. Prove each of the following set equalities both by Venn Diagram and by algebraic method.

(a) A - (B $\cap$ C) = (A - B) $\cup$ (A - C)

(b) A - (B $\cup$ C) = (A - B) $\cap$ (A - C)

(c) A $\cap$ (B - C) = (A $\cap$ B) - C = (A $\cap$ B) - (A $\cap$ C)

Hint: To prove the last form, use the equality A   $\cap$ C' = A   $\cap$ (A' $\cup$ C').

(d) A $\cup$ (B - C) = (A $\cup$ B) $\cap$ (A $\cup$ C') = (A $\cup$ B) - (A' $\cap$ C)

Answer 1

1. A - (B  C) = (A - B)  (A - C)

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1. A - (B  C) = (A - B)  (A - C)

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(c ) A  (B - C) = (A  B) - C = (A  B) - (A  C)

A  (B - C) =

(A  B) - C =

(A  B) - (A  C)

(d)A  (B - C) = (A  B)  (A  C') = (A  B) - (A'  C)

A  (B - C) =

(A  B)  (A  C') =

(A  B) - (A'  C)

1..

X-Y=XnY'
(XnY)'=X'uY'
A-(BnC)=An(BnC)'=An(B'uC')=(AnB')u(AnC')=(A-B)u(A-C)

2..

Use following properties:
X-Y=XnY'
(XnY)'=X'uY'
A-(BnC)=An(BnC)'=An(B'uC')=(AnB')u(AnC')=(A-B)u(A-C)