   Chapter 1.2, Problem 24E

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# Let f : A → B , where A and B are nonempty.Prove that f ( S 1 ∪ S 2 ) = f ( S 1 ) ∪ f ( S 2 ) for all subsets S 1 and S 2 of A .Prove that f ( S 1 ∩ S 2 ) ⊆ f ( S 1 ) ∩ f ( S 2 ) for all subsets S 1 and S 2 of A .Give an example where there are subsets S 1 and S 2 of A such that f ( S 1 ∩ S 2 ) ≠ f ( S 1 ) ∩ f ( S 2 ) .Prove that f ( S 1 ) − f ( S 2 ) ⊆ f ( S 1 − S 2 ) for all subsets S 1 and S 2 of A .Give an example where there are subsets S 1 and S 2 of A such that f ( S 1 ) − f ( S 2 ) ≠ f ( S 1 − S 2 ) .

(a)

To determine

To prove: f(S1S2)=f(S1)f(S2) for all subset of S1 and S2 of A. Here f:AB where A and B are nonempty set.

Explanation

Proof:

Suppose bf(S1S2).

Then there exists a x in S1S2 such that f(x)=b.

If x is in S1, then bf(S1).

If x is in S2, then bf(S2).

Those are the only choices so bf(S1)f(S2).

Hence, f(S1S2)f(S1)f(S2).

Now, suppose bf(S1)f(S2). Then bf(S1) or bf(S2)

(b)

To determine

To prove: f(S1S2)f(S1)f(S2) for all subsets S1andS2 of A. Here f:AB where A and B are nonempty set.

(c)

To determine

An example where there are subsets S1 and S2 of A such that f(S1S2)f(S1)f(S2). Here f:AB where A and B are nonempty set.

(d)

To determine

To prove: f(S1)f(S2)f(S1S2) for all subsets S1 and S2 of A .Here f:AB where A and B are nonempty set.

(e)

To determine

An example where there are subsets S1 and S2 of A such that f(S1)f(S2)f(S1S2). Here f:AB where A and B are nonempty set.

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