Q4. Show that the following four conditions are equivalent :

(i) A \subset B(ii) A – B = \phi (iii) A \cup B = B (iv) A \cap B = A

 

Answers (1)

First, we need  to show  A\subset B  \Leftrightarrow  A – B = \phi

Let  A \subset B 

To prove : A – B = \phi

Suppose  A – B \not = \equal \phi

this means, x \in A and x \not = B , which is not possible as  A \subset B .

SO,   A – B = \phi.

Hence, A \subset B \implies A – B = \phi.

Now, let A – B = \phi

To prove :  A \subset B 

Suppose, x \in A

 A – B = \phi so x \in B 

Since,  x \in A    and   x \in B  and A – B = \phi  so  A \subset B 

 Hence, A\subset B  \Leftrightarrow  A – B = \phi.

 

Let    A\subset B

To prove :  A \cup B = B

We can say B \subset  A \cup B

Suppose, x \in  A \cup B

means     x \in A    or   x \in B

If  x \in  A  

since   A\subset B so x \in B

Hence, A \cup B = B

and If  x \in B then also A \cup B = B.

 

Now, let  A \cup B =  B
To prove :   A\subset B

Suppose : x \inA

                A  \subset   A \cup B  so  x \in  A \cup B

                 A \cup B =  B   so x \in B

Hence,A\subset B

ALSO,    A\subset B  \Leftrightarrow   A \cup B =  B

 

NOW, we need to show A \subset B  \Leftrightarrow  A \cap B = A

Let  A \subset B

To prove :   A \cap B = A

Suppose : x \in A

We know  A \cap B \subset A    

                  x \in A \cap B  Also ,A  \subset A \cap B

           Hence, A \cap B = A

  

Let  A \cap B = A

To prove :   A \subset B

Suppose : x \in A

                 x \in  A \cap B      ( replacing A by  A \cap B )

                x \in A     and    x \in B

            \therefore  A \subset B

     A \subset B  \Leftrightarrow  A \cap B = A

 

 

 

 

 

 

 

 

 

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