question_answer

Which one of the following is correct?

A) \[A\cup (B-C)=A\cap (B\cap C')\]

B) \[A-(B\cup C)=(A\cap B')\cap C'\]

C) \[A-(B\cap C)=(A\cap B')\cap C\]

D) \[A\cap (B-C)=(A\cap B)\cap C\]

Correct Answer: B

Solution :

[b]Let a Venn-diagram be drawn taking three intersecting sets A, B and C under a universal set U. This makes 8 regions a to h as shown.

A has region a, b, d, e

B has region b, c, e, f

C has regions d, e, f, g

C? has regions a, b, c, h

B? has regions a, b, g, h

Statement : \[A\cup (B-C)=A\cap (B\cap C)\]

\[LHS\equiv (a,b,e,d)\cup b,c\equiv a,b,c,d,e.\]

\[RHS\equiv a,b,d,e\cap e,f\equiv e,\]

So, statement [a]is not correct.

Statement : \[A-(B\cup C)=(A\cap B')\cap C'\]

\[LHS\equiv (a,b,d,e)-(b,c,d,e,f,g)\equiv a.\]

\[RHS\equiv (a,b,d,e\cap a,d,g,h)\cap (a,b,c,h)\equiv a,\]

So, statement [b]is correct.

Correct statement is:

\[A-(B\cup C)=(A\cap B')\cap C'\]