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Math Help - Induction and Set Operations(Thinking)

  1. #1
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    Induction and Set Operations(Thinking)

    Hey I missed a few days, could anyone help me solve this proof?

    Prove that A1, A2, . . . ., An and B are sets, then (A1 ∩ A2 ∩ . . . ∩An) U B = (A1 U B) ∩ (A2 U B) ∩ . . . ∩(An U B).

    This comes from the Induction and Recursion chapter of Mathematical Induction.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Saint22 View Post
    Hey I missed a few days, could anyone help me solve this proof?

    Prove that A1, A2, . . . ., An and B are sets, then (A1 ∩ A2 ∩ . . . ∩An) U B = (A1 U B) ∩ (A2 U B) ∩ . . . ∩(An U B).

    This comes from the Induction and Recursion chapter of Mathematical Induction.
    recall the distributive law for sets: A \cup (B \cap C) = (A \cup B) \cap (A \cup C).

    Now, let P(n): (A_1 \cap A_2 \cap \cdots \cap A_n) \cup B = (A_1 \cup B) \cap (A_2 \cup B) \cap \cdots \cap (A_n \cup B) for all n \in \mathbb{N}, ~n \ge 1.

    P(1) is trivially true.

    Assume P(n) is true. We show P(n + 1)

    Define C = A_1 \cap A_2 \cap \cdots \cap A_n

    then, (A_1 \cap A_2 \cap \cdots \cap A_n \cap A_{n + 1}) \cup B = (C \cap A_{n + 1}) \cup B = (C \cup B) \cap (A_{n + 1} \cup B) By the distributive law.

    since P(n) is true. (C \cup B) = (A_1 \cup B) \cap (A_2 \cup B) \cap \cdots (A_n \cup B)

    so that we have (A_1 \cap A_2 \cap \cdots \cap A_n \cap A_{n + 1}) \cup B = (A_1 \cup B) \cap (A_2 \cup B) \cap \cdots \cap (A_{n + 1} \cup B)

    so that P(n + 1) is true.

    This completes the inductive proof
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