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29 November, 06:23

Show that (A | B) U (B / A) = (AUB) / (B n A).

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  1. 29 November, 06:34
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    We have to prove,

    (A / B) ∪ (B / A) = (A U B) / (B ∩ A).

    Suppose,

    x ∈ (A / B) ∪ (B / A), where x is an arbitrary,

    ⇒ x ∈ A / B or x ∈ B / A

    ⇒ x ∈ A and x ∉ B or x ∈ B and x ∉ A

    ⇒ x ∈ A or x ∈ B and x ∉ B and x ∉ A

    ⇒ x ∈ A ∪ B and x ∉ B ∩ A

    ⇒ x ∈ (A ∪ B) / (B ∩ A)

    Conversely,

    Suppose,

    y ∈ (A ∪ B) / (B ∩ A), where, y is an arbitrary.

    ⇒ y ∈ A ∪ B and x ∉ B ∩ A

    ⇒ y ∈ A or y ∈ B and y ∉ B or y ∉ A

    ⇒ y ∈ A and y ∉ B or y ∈ B and y ∉ A

    ⇒ y ∈ A / B or y ∈ B / A

    ⇒ y ∈ (A / B) ∪ (B / A)

    Hence, proved ...
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