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 x^{2} < y^{2}.

1 | Suppose 0 < x < y. |

2 | Since 0 < x, multiplying both sides of the inequality by x we get 0 < x^{2} < xy. |

3 | Similarly, since 0 < y, multiplying both sides of the inequality by y we get 0 < xy < y^{2}. |

4 | Thus, x^{2} < xy and xy < y^{2} which implies that x^{2} < y^{2}. |

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. |