Each of the following is an assertion for all subsets A and

Each of the following is an assertion for all subsets A and B or every (universal) set U. ( A\' denotes the complement of A.) For each of them decide whether it is true for all possible A, B, and U, or whether it may be false. Give a proof if it is always true. Construct a counterexample if it can be false.

Solution

a) This is true

Let P = (A U B)\' and Q = A\' B\'

Let x be an arbitrary element of P then x P x (A U B)\'

x (A U B)

x A and x B

x A\' and x B\'

x A\' B\'

x Q

Therefore, P Q …………….. (i)

Again, let y be an arbitrary element of Q then y Q y A\' B\'

y A\' and y B\'

y A and y B

y (A U B)

y (A U B)\'

y P

Therefore, Q P …………….. (ii)

Now combine (i) and (ii) we get; P = Q i.e. (A U B)\' = A\' B\'

b) A B means that every element of A is also an element of B.

So, A U (AB) B is false.

Example: Let A={1,2,3,4}, B={3,5,6}

                        AB ={3}

                        A U (AB) ={1,2,3,4}

But every element of A U (AB) is not an element of B.