Consider the statement. For all sets A, B, and C, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). Fill in the blanks in the following proof for the stateme

Question

Consider the statement. For all sets A, B, and C, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). Fill in the blanks in the following proof for the statement. (In the proof, let ∩ and ∪ stand for the words “intersection” and “union,” respectively.) Proof: Suppose A, B, and C are any sets. [To show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), we must show that A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C) and that (A ∩ B) ∪ (A ∩ C) ⊆ A ∩ (B ∪ C).]
Proof that A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ (A ∩ C): Let x ∈ A ∩ (B ∪ C).

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Luna 3 weeks 2021-09-10T18:49:09+00:00 1 Answer 0

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    2021-09-10T18:51:06+00:00

    Answer:

    By definition it follows that

    x \in (A \cap B)  \cup(A \cap C)\\

    Step-by-step explanation:

    \text{if} \,\,\,\,\, x \in A \cap (B \cup C)

    by definition

    x\in A      \,\,\,\,\,\,\,\, \text{and}          \,\,\,\,\,\,\,\, x \in (B \cup C).    \\ \text{therefore}  \,\,\,\,\,x\in A \,\,\,\,\, \text{and} \,\,\,\,\, x\in B \,\,\,\,\, \text{or}  \,\,\,\,\,\, x\in \text{C}

    Then it follows that

    x \in (A \cap B)  \cup(A \cap C)\\

    The other side is pretty much the same.

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