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# Examples of contrapositives conjectures, & counterexamples

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1. For each of the following statements, write the contrapositive statement, and prove the
original statement by proving its contrapositive:

(a) If m^2 + n^2 &#8800; 0, then m &#8800; 0 or n &#8800; 0.

2. What is wrong with the following proof of the conjecture "If n^2 is positive, then n is positive.":

Proof: Suppose that n^2 is positive. Because the conditional statement "If n is positive, then n^2 is positive" is true, we can conclude that n is positive.

3. A rational number is a number that can be expressed as the ratio of two integers p and
q such that q &#8800; 0.

(a) Prove that the product of any two rational numbers x and y is a rational number.

4. The term modulus (denoted mod or %) is used to describe the remainder when one integer
is divided by another. Thus, we write a mod b = r to mean that r is the remainder when a is divided
by b. Provide a counterexample to each of the following statements about integers that is false:
(a) If (a mod b) = (b mod c), then a = b.

(b) If (a mod b) = c, then ((a + 1) mod b) = c + 1

##### Solution Summary

Four problems are worked out to find the contrapositive of a given statement, determine if the contrapositive of a statement is true, prove that the product of rational numbers is rational, and prove examples using modular arithmetic.

##### Solution Preview

1. For each of the following statements, write the contrapositive statement, and prove the
original statement by proving its contrapositive:

(a) If m^2 + n^2 &#8800; 0, then m &#8800; 0 or n &#8800; 0.

Contrapositive: If m = 0 and n = 0, then m2 + n2 = 0.
Proof: If m = 0, then m^2 = 0 * 0 = 0. If n = 0, then n^2 = 0 * 0 = 0. Therefore m^2 + n^2 = 0 + 0 = 0. Since the contrapositive shares the same truth value as the original statement, then the original statement is also true since the contrapositive is true.

2. What is wrong with the following proof of the conjecture "If n^2 is positive, then n is positive.": ...

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