Proofs on Sets: Help with Proving (A \cup B) X C

[bigrodey77]

Hello all,

I'm having a hard time trying to prove a few things. I'm looking for a little help because I cannot seem to grasp the concept of proofs and what constitutes a valid proof and if my proof is wrong, correcting it.

I have a proof done and if anyone could "critique" it I would be very grateful.

Prove: (A \cup B) X C = (A X C) \cup (B X C)

Proof:
Let x \in (A \cup B) X C
Then x is of the type (y,z) where y \in A and z \in C
Then y \in A or y \in B
Since z \in C, (y,z) \in A X C or
Since z \in C, (y,z) \in B X C
Then (y,z) \in (A X C) \cup (B X C)
Therefore (A \cup B) X C = (A X C) \cup (B X C)

Thanks for your time,

Ryan

 

[EnumaElish]

What is X? does it stand for \cap?

 

[d_leet]

What is X? does it stand for \cap?

I would assume that it represents the cartesian product.

 

[EnumaElish]

I would assume that it represents the cartesian product.

"Duh!"

Let x in (A U B) X C
Then x is of the type (y,z) where y in A or B and z in C.

Otherwise your logic is correct.

 

[bigrodey77]

Thank you guys very much for your responses, I am sure I'll have a couple more here tomorrow...

Thanks again!

 

[HallsofIvy]

Technical point: it would be better to say IF x\in (A\Cup B)\cross C. "let x ..." runs into trouble if the set is empty!

More important point: you have proved that (A\Cup B) X C \subset (A X C)\Cup (B X C), not that they are equal you still have to prove that "if x is in (A X C)\Cup (B X C), then it is in (A\Cup B) X C[\itex].