Theorems on Relations and Functions

Theorem 1:

Statement:

Proof:

Let (a,b) be an arbitrary element of A x B.

Theorem 2:

Statement:

Proof:

Theorem 3:

Statement:

Proof:

Hence the theorem.

Theorem 4:

Statement:

Proof:

Hence the theorem.

Theorem 5:

Statement:

Proof:

Similarly it can be proved that A x (B - C) Ì (A x C) - (B x C)

Hence the theorem.

Theorem 6:

Statement:

If A, B, C be any three sets then

Proof:

Hence the theorem .

Similarly it can be proved that (A x B)' Ì (A' x B) È (A x B') È (A' x B')

Theorem 7:

Statement:

If A, B and C are any three sets, prove that

Proof:

---- (1)

From (1) and (2) we get,

Theorem 8:

Statement:

Proof:

Theorem 9:

Statement:

Proof:

.… (1)

From (1) and (2) we get,

Theorem 10:

Statement:

If A and B are non-empty sets. Prove that

A x B = B x A, if and only if A = B.

Proof:

Let A = B

Then we have to prove that A x B = B x A.

Again, let A x B = B x A, then we have to show that A = B.

Let y be any element of B.

By definition of equality A = B.