Quiz 2

Justify all answers with neat and organized work. Clearly indicate your answers. 20 points possible.

1. (5 pts.) Consider the following statement.

"n Î Z, if n is prime, then n is odd or n=2. (*)

(a) Write the contrapositive of statement (*).

(b) Write the converse of statement (*).

(c) Write the inverse of statement (*).

(d) Write the negation of statement (*).

2. (5 pts.) State whether the given argument has a valid or invalid form. (You do not have to justify your answers.)

(a)

All freshmen must take writing.

      Therefore,

Caroline must take writing.

(b)

All healthy people eat an apple a day.

      Therefore,

Herbert does not eat an apple a day.

(c)

If a product of two numbers is 0, then at least one of the numbers is 0.

      Therefore,

Neither (x-1) nor (x+1) equals 0.

(d)

All cheaters sit in the back row.

      Therefore,

George is a cheater.

(e)

All honest people pay their taxes.

      Therefore,

Darth does not pay his taxes.

3. (5 pts.) Prove this theorem.

Theorem The product of any even integer and any odd integer is even.

4. (5 pts.) Prove this theorem.

Theorem For all integers a, b, and c, if a | b and a | c, then a | (b-c).