Question

Hello, can you please solve 21.11, using the Theorem 21.13? Thank you.

Problem 21.11. Prove the following corollary of Theorem 21.13 above. Theorem 21.13. Let A, B,C, and D be nonempty sets with AC and Bn D. Then

Answer 1

Defination : we say that two set $A \approx B$ if there exist a bijective map .

f:A→B

Given , $A \approx C , B \approx D$ , so there exist two map
f:A→C
and $g: B\rightarrow D$ which are bijective .

Now , $h: A\times B\rightarrow C\times D$ 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)

$\Rightarrow$ ( f(x) , g(y)) = ( f(a) , g(b))

$\Rightarrow$ f (x) = f(a) and g(y) = g(b)

$\Rightarrow$ x = a and y = b , since f , g are one to one

$\Rightarrow$ ( x , y ) = ( a , b)

So h is one - to-one .

Onto : Let (a , b) $\in$ C ×D

$\Rightarrow$$a \in C , b \in D$

As f and g are onto there exist $x\in A , y \in B$ such that

f(x) = a and g(x) = b

$\Rightarrow$ h (x , y ) = ( f(x) , g(y)) = ( a , b)

$\Rightarrow$ ( a , b) has a prem age under h .

So h is onto .

And consequently h is bijection .

Hence ,  $A \times B \approx C \times D$ .

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Please comment if needed.