# Q4. Show that the following four conditions are equivalent :(i) A  B(ii) A – B =  (iii) A  B = B (iv) A  B = A

First, we need  to show  A B    A – B =

Let  A  B

To prove : A – B =

Suppose  A – B

this means, x  A and x  B , which is not possible as  A  B .

SO,   A – B = .

Hence, A  B  A – B = .

Now, let A – B =

To prove :  A  B

Suppose, x  A

A – B =  so x  B

Since,  x  A    and   x  B  and A – B =   so  A  B

Hence, A B    A – B = .

Let    A B

To prove :  A  B = B

We can say B   A  B

Suppose, x   A  B

means     x  A    or   x  B

If  x   A

since   A B so x  B

Hence, A  B = B

and If  x  B then also A  B = B.

Now, let  A  B =  B
To prove :   A B

Suppose : x A

A     A  B  so  x   A  B

A  B =  B   so x  B

Hence,A B

ALSO,    A B     A  B =  B

NOW, we need to show A  B    A  B = A

Let  A  B

To prove :   A  B = A

Suppose : x  A

We know  A  B  A

x  A  B  Also ,A   A  B

Hence, A  B = A

Let  A  B = A

To prove :   A  B

Suppose : x  A

x   A  B      ( replacing A by  A  B )

x  A     and    x  B

A  B

A  B    A  B = A

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