# 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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