Defination : we say that two set if there exist a bijective map .
Given , , so there exist two map and which are bijective .
Now , be defined by ,
h ( x , y ) = ( f(x) , f(y) )
Now we prove that h is a bijection .
one - to - one :
Let h(x , y ) = h ( a , b)
( f(x) , g(y)) = ( f(a) , g(b))
f (x) = f(a) and g(y) = g(b)
x = a and y = b , since f , g are one to one
( x , y ) = ( a , b)
So h is one - to-one .
Onto : Let (a , b) C ×D
As f and g are onto there exist such that
f(x) = a and g(x) = b
h (x , y ) = ( f(x) , g(y)) = ( a , b)
( a , b) has a prem age under h .
So h is onto .
And consequently h is bijection .
Hence , .
.
.
.
Please comment if needed.
Problem 21.11. Prove the following corollary of Theorem 21.13 above. Theorem 21.13. Let A, B,C,...
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