Relations and Functions


   
 
Some Important Theorems
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.
 
  
     
   
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