Question

1.)Which of the expressions is equivalent to the following statement:

The sum of two even numbers is even.

a.)	If x is even or y is even, then x + y

b.)	If x is even or y is even, then x + y is even

c.)	If x is even and y is even, then x + y is not even.

d.)	If x is even and y is even, then x + y is even

2.) Find a counterexample for the statement: every month of the year has 30 or 31 days.

a.)

March

b.)

December

c.)

January

d.)

February

3.) If direct proof is used to prove the following statement: If x is a real number and x ≤ 3, then 12 - 7x + x*x ≥ 0. What is the hypothesis?

a.)	If x is not a real number or x > 3

b.)	If x is a real number and x ≤ 3

c.)	12 - 7x + x*x < 0

d.)	12 - 7x + x*x ≥ 0

4.) If proof by contrapositive is used to prove the following statement: For every real number x, if x is irrational, then -x is also irrational. What is the hypothesis?

a.)	x is rational.

b.)	x is irrational.

c.)	-x is rational.

d.)	-x is irrational.

5.) Consider the following statement: For every positive integer n less than 3, n + n ≥ n2. Which facts must be checked in a proof by exhaustion of the statement?

a.)	n = 0, 0 + 0 = 0; n = 1, 1 + 1 > 1

b.)	n = 1, 1 + 1 > 1; n = 2, 2 + 2 = 22

c.)	n = 0, 0 + 0 = 0; n = 1, 1 + 1 > 1; n = 2, 2 + 2 = 22

Answer 1

1(d) If x is even and y is even, then x + y is even

Explanation: BOTH should be even for the sum to be even

2(d) February
Explanation: As we know it can have 28/29 days


3)b.  If x is a real number and x ≤ 3

4)a. x is rational.

5) c. n = 0, 0 + 0 = 0; n = 1, 1 + 1 > 1; n = 2, 2 + 2 = 22


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