# Thread: Cartesian Products: Proof of Equality of Sets

1. ## Cartesian Products: Proof of Equality of Sets

Problem:

If S x T = U x V, then show that S = U and T = V.

Incomplete Solution:

Step 1) S x T = {(a,b) | a e S, b e T}; U x V = {(c,d) | c e U, d e V}

Step 2) {(a,b) | a e S, b e T} = {(c,d) | c e U, d e V}

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Step n-1) a = c for all a e S, c e U; b = d for all b e T, d e V

Step n) S = U; T = V

Question: How do I make a formal connection between steps 2 and N-1?

2. Originally Posted by gryphon_gold
Problem:

If S x T = U x V, then show that S = U and T = V.
Assume
$\displaystyle S\times T=U\times V$ and $\displaystyle S=U \mbox{ and }T=V$ is false. Then, be de Morgan laws. $\displaystyle S\not =U \mbox{ or } T\not = V$
If,
$\displaystyle S\not = U$ then $\displaystyle \exists [a\in S \mbox{ and }a\not \in U] \mbox{ or }[a\not \in S \mbox{ and }a\in U]$
Then,
$\displaystyle \{(s,t)|a\in S \mbox{ and }t\in T\}\not = \{(u,v)|u \in U\mbox{ and } v\in V\}$
$\displaystyle S=U \mbox{ and } T=V$