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Math Help - Set Theory - Element Argument

  1. #1
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    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.
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  2. #2
    Senior Member Danneedshelp's Avatar
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    First, assume A⊆B.  Now, suppose x∈A∪C.  Then x∈A or x∈C<br />
case 1:  If x∈A, then x∈B because A⊆B.  Thus, x∈B or x∈C.<br />
case 2:  If x∈C, then x∈B or x∈C.<br />
So, either way, x∈B∪C.  Therefore, A∪C⊆B∪C.    ☐<br /> <br />
Remember this definition:<br /> <br />
A⊆B ⇔ ∀x(x∈A⇒x∈B)
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  3. #3
    Senior Member Danneedshelp's Avatar
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    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?
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  4. #4
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    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.
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