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 [itex](A\Cup B) X C[\itex].