Prove That (A∪B) ∩ (A∪B’) = A
Question:
Using properties of sets, show that for any two sets \( A \) and \( B \),
\[ (A\cup B)\cap(A\cup B’)=A \]Solution
Consider the left-hand side:
\[ (A\cup B)\cap(A\cup B’) \]Using the distributive law,
\[ =(A\cup(B\cap B’)) \]Now,
\[ B\cap B’=\phi \]Therefore,
\[ =A\cup\phi \]Using the identity law,
\[ A\cup\phi=A \]Hence,
\[ (A\cup B)\cap(A\cup B’)=A \]Hence proved.