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037 G
Lv 6
037 G asked in Science & MathematicsMathematics · 1 decade ago

Logical proof Anyone?

Given

(A ∩ B) = (A ∩ C)

and

(A ∪ C) ⊆ (A ∪ B)

Prove

C⊆ B

1 Answer

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  • ?
    1 decade ago
    Favorite Answer

    This result can be proved by contradiction.

    Suppose C is not a subset of B, so that there exists an element x in C that does not belong to B.

    Observe that x does not belong to A, because then x would belong to A ∩ C but not belong to A ∩ B; this would contradict (A ∩ B) = (A ∩ C).

    Then x is an element of A ∪ C (because it is an element of x), but it is not an element of A ∪ B (because it does not belong to A or B). However this contradicts the fact that A ∪ C is a subset of A ∪ B.

    Source(s): Mathematics professor
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