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Math 2803–HEH: Foundations of Mathematical Proof
Homework 2
Spring 2015
due January 23rd
There are 9 problems on 2 pages of this homework assignment.
You may assume that sums and products of integers are integers, and that an integer is even if and
only if it is not odd. You may use anything that has already been proved in class or in webwork.
Any other results you would like to use must be proved first by you in this assignment.
1. Prove the following from the definition of even/odd integers by the desired method.
(a) Direct proof: For all integers n, if n is odd, then 9n + 5 is even.
(b) Contrapositive proof: For all integers n, if 7n + 5 is odd, then n is even.
(c) Proof by contradiction: For all integers n, if 3n + 2 is even, then n is even.
2. Disprove the following universal statements by finding a counterexample. Justify your answer.
(a) For every odd positive integer n, 3 divides n2 − 1.
(b) Let n ∈ Z. If n2 + 3n is even, then n is odd.
(c) If n3 + n is even for an integer n, then n is even.
3. Prove the following existential statements by giving an example. Justify your answer.
(a) There exists an integer whose cube equals its square.
(b) There exist odd integers a, b, and c such that a + b + c = 1.
(c) There exists an integer which is greater than e but less than π.
4. Disprove the following existential statements by proving a related universal statement.
(a) There exists an odd integer n such that n2 + 2n + 3 is odd.
(b) There exists an integer n such that n3 − n + 1 is even.
(c) There exists an integer n such that 3n2 − 5n + 1 is an even integer.
5. Let a ∈ Z. Prove that 3 | a2 if and only if 3 | a. You may use the fact that every integer can
be written as exactly one of 3k, 3k + 1, or 3k + 2 for some integer k.
√
6. Prove that 3 is irrational. Hint: use Problem 5.
7. Prove or disprove.
(a) The sum of two rational numbers is rational.
(b) The sum of an irrational number and a rational number is irrational.
(c) The sum of two irrational numbers is irrational.
(d) The product of two rational numbers is rational.
(e) The product of an irrational number and a rational number is irrational.
(f) The product of two irrational numbers is irrational.
8. Let a be an irrational number, s a real number, and r a nonzero rational number. Prove,
directly from the definition, that either ar + s or ar − s is irrational.
9. Definition: An integer n divides an integer m, denoted n | m, if (and only if) there exist an
integer k such that m = nk. Otherwise, n - m.
Let a, b, c, d, x and y be integers with a 6= 0 and b 6= 0.
(a) If a | c and b | d, then ab | cd.
(b) If a | c and a | d, then a | cx + dy.
(c) If a | b and b | a, then a = b or a = −b.
(d) If a - cd, then a - c and a - d.
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