Set Theory - Element Argument

• Apr 22nd 2009, 04:14 PM
MathWise
Set 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 PM
Danneedshelp
\$\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 PM
Danneedshelp
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 PM
MathWise
re
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.