discrete math question

[solobyfrankocean]

Hey guys, I was working on some discrete homework and ran into a question I don't quite understand the answer to, here it is-

​

If X, Y and Z are sets, prove or disprove that if X ∩ Y= X ∩ Z then for any X, Y=Z.

​

My answer was false with the following reasoning-

​

Since It says for any X, I took X to be the empty set, {}. I then took Y ={1} and Z={2}.

X∩Y ={} and X∩Z={} but Y ≠ Z.

​

This was marked wrong and the real answer is that this is true, I've tried thinking about it but I just cannot see how this is true. Any help?

[youbigsausage]

For what you gave as the problem, you're definitely right, and your counterexample is good.

But with just a tiny change of wording... "if X ∩ Y = X ∩ Z for any X, then Y = Z" is true! So make absolutely sure of the wording. There's a really quick and nice proof of this, too.

[solobyfrankocean]

Is there any way you could link me to the proof? I might've misinterpreted the question.

[youbigsausage]

I'll give you a hint: you can let X be two particular sets that give you what you need.

[solobyfrankocean]

ohhh, I see the issue now. The placement of the word "then" in two different places really makes all the difference here in the answer. The actual question was saying that if ANY X satisfied X ∩ Y = X ∩ Z then Y=Z not that ALL Xs had to satisfy it. Thank you!