In Exercises, verify Theorem for the given sets.TheoremLet a, b, and c be integers.(a) If...

In Exercises, verify Theorem for the given sets.

Theorem

Let a, b, and c be integers.

(a) If a | b and a | c, then a | (b + c).

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(b) If a | b and a | c, where b >c, then a | (b − c).

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(c) If a | b or a | c, then a | bc.

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(d) If a | b and b | c, then a | c.

Proof

(a)If a | b and a | c, then b = k1a and c = k2a for integers k1 and k2. So b + c = (k1 + k2)a and a | (b + c).

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(b) This can be proved in exactly the same way as (a).

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(c) As in (a), we have b = k1a or c = k2a. Then either bc = k1ac or bc = k2ab, so in either case bc is a multiple of a and a | bc.

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(d) If a | b and b | c, we have b = k1a and c = k2b, so c = k2b = k2(k1a) = (k2k1)a and hence a | c.

Exercise

(a) A = {a, b, c, d, e, f}, B = {a, c, f, g, h, i, r}

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(b) A = {a, b, c, d, e}, B = {f, g, r, s, t, u}