Thread: 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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.
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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?

Originally Posted by gryphon_gold

Problem:

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

Assume
and is false. Then, be de Morgan laws.
If,
then
Then,

Contradiction,
Thus,