Question 1: For any two sets and , prove that :

Answer:

To show:

We show that and vice versa

Let and

and [ Since and ]

and

This is true for all . Hence

Conversely, let

and

and [ Since and ]

This is true for all

Hence .

Therefore . Hence proved.

Question 2: For any two sets and , prove the following :

(i) (ii) (iii) (iv)

Answer:

(i) To prove:

LHS

[ Since Intersection distributes over union ]

[ Since ]

[ Since for any set ]

RHS. Hence proved.

(ii) To prove:

For any set and , we have De-morgan’s Laws.

LHS

[ By De-morgans law]

[ Since ]

[ Since ]

[ Since for any set ]

RHS. Hence proved.

(iii) To prove:

LHS

[ By De-morgans law]

[ Since ]

RHS.

(iv) To prove:

RHS

[ Since ]

[ Since ]

[ By De-morgans law and associative law ]

[ Since intersection distributes over union and ]

[Since ]

[Since ]

LHS

Question 3: If are three sets such that , then prove that

Answer:

We have sets and

To show

Let and and

and [ Since ]

Thus,

This is true for all

Therefore

Question 4: For any two sets and , prove that,

(i) (ii) (iii) (iv) (v)

Answer:

i) To prove:

ii) To prove:

iii) To prove:

Let

and

and

Therefore

iv) To prove:

Let or

or and

[ Since ]

This is true for all

Therefore … … … … … i)

Conversely,

Let

and

or [ Since ]

Therefore … … … … … ii)

From i) and ii) we get

v) To prove:

Let

The either or

Therefore … … … … … i)

Conversely,

Let

or

and or and

Therefore … … … … … ii)

From i) and ii) we get