By contradiction, prove that for all sets A, B, and C, (A − C) ∩ (B − C) ∩ (A − B) = ∅
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statement is true, and here is a proof:
Proof: Suppose A, B, and C are sets such that A−B⊆A−C. We want to prove that A∩C=∅ by contradiction.
Suppose A∩C≠∅. That is there is x∈A∩C. Also, since A−B⊆A−C, suppose there is a x∈A−B, then x∈A−C. That is, x∉C. Therefore, that implies that x∉A∩C which contradicts with given that x∈A∩C. Therefore, concluding by contradiction: A∩C=∅. End of proof.
Is this proof correct? Any issues?.
Proof: Suppose A, B, and C are sets such that A−B⊆A−C. We want to prove that A∩C=∅ by contradiction.
Suppose A∩C≠∅. That is there is x∈A∩C. Also, since A−B⊆A−C, suppose there is a x∈A−B, then x∈A−C. That is, x∉C. Therefore, that implies that x∉A∩C which contradicts with given that x∈A∩C. Therefore, concluding by contradiction: A∩C=∅. End of proof.
Is this proof correct? Any issues?.
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