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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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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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