Spring 2005

Be neat and organized. Clearly indicate your answers. 20 points possible.

1. (5 pts.) The Zero Product Property says that if a product of two real numbers is 0, then one of the numbers must be 0.

(a) Write this property formally using quantifiers (" and/or $) and variables.

(b) Write the contrapositive of your answer to part (a).

2. (5 pts.) Fill in the blanks in the proof.

Theorem. The sum of any two even integers is even.

Proof. Suppose k and w are particular but arbitrarily chosen (a). [We must show that (b).] By definition of even, k=2n and w=2m for some (c). By substitution and algebra, k+w=(d)=2(n+m). Let j=n+m. Note that j is an integer because (e). Hence k+w=2j where j is an integer. It follows by definition of even that k+w is even. \blacksquare

(a)

(b)

(c)

(d)

(e)

3. (5 pts.) Prove the given statement directly from the definitions.

The sum of any integer and any rational number is a rational number.

4. (5 pts.) Prove the given statement directly from the definitions.

For all integers b, k, and n, if b | k and k | n, then b | (8k-5n).