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WS-11-16-2010

# WS-11-16-2010 - B(ii f X ∪ X = f X ∪ f X(iii f X ∩ X...

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In the following problems, A and B are nonempty sets, and f : A B is an arbitrary function from A to B . Let U and V be arbitrary subsets of B . Prove the following identities: 1. (a) f - 1 ( B ) = A . (b) f - 1 ( U V ) = f - 1 ( U ) f - 1 ( V ). (c) f - 1 ( U V ) = f - 1 ( U ) f - 1 ( V ). (d) f - 1 ( U - V ) = f - 1 ( U ) - f - 1 ( V ). Let X and X be arbitrary subsets of A . Prove the following relations:
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Unformatted text preview: B . (ii) f ( X ∪ X ) = f ( X ) ∪ f ( X ). (iii) f ( X ∩ X ) ⊆ f ( X ) ∩ f ( X ). Show by example that equality need not hold. (iv) f ( X-X ) ⊇ f ( X )-f ( X ). Show by example that equality need not hold....
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