Question 4: Put the following statements in order to prove that if 3n+4 is even then n is even. Put N next to the statements that should not be used.

7	Thus, if n is odd then 3n+4 is odd or by contraposition, if 3n+4 is even then n is even.
6	Therefore, by definition of odd, 3n+4 is odd.
1	Suppose n is odd.
2	By definition of odd, there exists integer k such that n=2k+1.
4	Since k is an integer, t = 3k + 1 is also an integer.
5	Thus, there exists an integer t such that 3n+4 = 2t + 1
N	Thus, if n is odd then 3n+4 is odd or by contradiction, if 3n+4 is even then n is even.
N	Suppose 3n+4 is even.
N	By definition of odd, n=2k+1.
3	Then, 3n+4 = 3(2k+1)+4 = 6k+7 = 2(3k+3) + 1.

Question 5: Put the following statements into order to prove that if both x and y are positive and x < y, then x² < y².

1	Suppose 0 < x < y.
2	Since 0 < x, multiplying both sides of the inequality by x we get 0 < x2  < xy.
3	Similarly, since 0 < y, multiplying both sides of the inequality by y we get 0 < xy < y2.
4	Thus, x2 < xy and xy < y2 which implies that x2 < y2.

Question 6: Prove that is x and y are rational then their product is also rational. First write a proof on your own and then put the following statements into order to obtain the proof. Put N next to the statements that should not be used in the proof.

N	Suppose x = 2/3 and y = 3/5.
3	Then, xy = ac / bd.
N	Then, xy = 2/5 which is rational.
N	By definition of rational, there exists a, b (not equal to 0), c and d (not equal to 0) such that x = a/b and y = c/d.
4	Since a, c are integers, e = ac is also an integer.
6	Thus, xy = e/f, and by definition of rational xy is rational.
N	Since the product of two fractions is a fraction xy is rational.
1	Suppose x and y are arbitrary rational numbers.
5	Similarly since b, d are non-zero integers, f = bd is also a non-zero integer.
N	By definition of rational, there exist real numbers a, b (not equal to 0), c and d (not equal to 0) such that x = a/b and y = c/d.
2	By definition of rational, there exist integers a, b (not equal to 0), c and d (not equal to 0) such that x = a/b and y = c/d.