Let A and B be sets. Then we denote the set of functions with domain A...

Let A and B be sets. Then we denote the set of functions with domain A and codomain B as B^A. In other words, an element f∈B^A is a function f:A→B.

Prove: Let ?,?∈?^? (that is, ? and ? are real-valued functions with domain ?) and define a relation ≡ on ?^? by ?≡?⟺?(0)=?(0). (That is, ? and ? are equivalent if and only if they share the same value at ?=0) Then ≡ is an equivalence relation on ?^?.

Prove: Suppose that ?:?→? f:A→B and ?,?⊆? Then ?(?∩?)⊆?(?)∩?(?)

Prove: Suppose that ?:?→? f:A→B and E,F⊆B. Then ?^(−1)(?)∪?(−1)(?)⊆?(?∪?)

Prove: Suppose that {Ai} is a partititon of B. Then the relation x∼y for defined x,y∈A by x∼y⟺∃Ai(x,y∈Ai) is an equivalence relation.

Expert Solution

I felt there were some typing errors at few places in question. I tried best possible way to rectify and solve them accordingly.

I hope it helps. Please feel free to revert back with further queries (if any) in comments.

Colby Messinger answered 5 months ago

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