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# Let A, B, C and D be sets such that | A| = |C| and |B| = |D|. Prove that |A × B| = |C × D|

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## Answer to a math question Let A, B, C and D be sets such that | A| = |C| and |B| = |D|. Prove that |A × B| = |C × D|

Darrell
4.5
To prove that |A × B| = |C × D|, we need to show that there exists a bijective function between the two sets.

Let's consider the function f: A × B → C × D defined as f$a, b$ = $c, d$ where c is any element in C and d is any element in D. Since |A| = |C| and |B| = |D|, we know that there exists a bijective function g: A → C and a bijective function h: B → D.

Now, let's define a function F: C × D → A × B as F$c, d$ = $g^(-1$$c$, h^$-1$$d$), where g^$-1$ and h^$-1$ are the inverse functions of g and h, respectively.

We will prove that both f and F are bijections.

First, let's show that f is injective. Suppose $a1, b1$ and $a2, b2$ are two elements in A × B such that f$a1, b1$ = f$a2, b2$. This implies that $g(a1$, h$b1$) = $g(a2$, h$b2$). Since g and h are both injective functions, we conclude that a1 = a2 and b1 = b2. Therefore, f is injective.

Next, let's show that f is surjective. Let $c, d$ be an element in C × D. Since g and h are both surjective functions, there exists a1 in A such that g$a1$ = c, and there exists b1 in B such that h$b1$ = d. Therefore, f$a1, b1$ = $c, d$. Hence, f is surjective.

Now, let's show that F is injective. Suppose $c1, d1$ and $c2, d2$ are two elements in C × D such that F$c1, d1$ = F$c2, d2$. This implies that $g^(-1$$c1$, h^$-1$$d1$) = $g^(-1$$c2$, h^$-1$$d2$). Since g^$-1$ and h^$-1$ are both injective functions, we conclude that c1 = c2 and d1 = d2. Therefore, F is injective.

Finally, let's show that F is surjective. Let $a, b$ be an element in A × B. Since g and h are both surjective functions, there exists c1 in C such that g$a$ = c1, and there exists d1 in D such that h$b$ = d1. Therefore, F$c1, d1$ = $g^(-1$$g(a$), h^$-1$$h(b$)) = $a, b$. Hence, F is surjective.

Since f is a bijection from A × B to C × D, and F is a bijection from C × D to A × B, we can conclude that |A × B| = |C × D|.

Answer: |A × B| = |C × D|

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