Element argument to prove that for all sets A, B, and C:

If A ⊆ B, then A U C ⊆ B U C.

please if you can help i cannot figure this one out.

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- Apr 22nd 2009, 04:14 PMMathWiseSet Theory - Element Argument
Element argument to prove that for all sets A, B, and C:

If A ⊆ B, then A U C ⊆ B U C.

please if you can help i cannot figure this one out. - Apr 22nd 2009, 05:01 PMDanneedshelp
$\displaystyle First, assume A⊆B. Now, suppose x∈A∪C. Then x∈A or x∈C

case 1: If x∈A, then x∈B because A⊆B. Thus, x∈B or x∈C.

case 2: If x∈C, then x∈B or x∈C.

So, either way, x∈B∪C. Therefore, A∪C⊆B∪C. ☐

Remember this definition:

A⊆B ⇔ ∀x(x∈A⇒x∈B)$ - Apr 22nd 2009, 05:06 PMDanneedshelp
First, assume A⊆B. Now, suppose x∈A∪C. Then x∈A or x∈C

case 1: If x∈A, then x∈B because A⊆B. Thus, x∈B or x∈C.

case 2: If x∈C, then x∈B or x∈C.

So, either way, x∈B∪C. Therefore, A∪C⊆B∪C. ☐

Remember this definition:

A⊆B ⇔ ∀x(x∈A⇒x∈B)

Do you understand how the definition was used to prove the above theorem? - Apr 22nd 2009, 05:21 PMMathWisere
Thank you very much, I was very close to that just did not break it into cases. I will learn that definition, thank you for the help.