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Chapter 2: Problem 15

15\. Provide a proof by contradiction for the following: For every integer \(n\), if \(n^{2}\) is odd, then \(n\) is odd.

Short Answer

Expert verified
The statement 'For every integer \(n\), if \(n^{2}\) is odd, then \(n\) is odd' is proven true via contradiction. By assuming the contrary and deriving a contradiction, the statement is proven to be true.

Step by step solution

01

Statement of Assumption

Since this is a proof by contradiction, begin by assuming the opposite of what is to be proved. Here, the statement to prove is that 'if \(n^{2}\) is odd then \(n\) is odd'. The contrary is 'there exists integer \(n\) such that \(n^{2}\) is odd but \(n\) is not odd, which means \(n\) is even.'
02

Definition of Even Number

Remember that by definition, an even number can be written as \(2k\), where \(k\) is an integer. So in line with the assumption, let \(n\) be equal to \(2k\).
03

Substitution into Equation

Substitute \(2k\) for \(n\) in the equation \(n^{2}\). Therefore, the equation becomes \((2k)^{2}\).
04

Simplification and Conclusion of Contradiction

This equation simplifies to \(4k^{2}\), which is clearly even because it's a multiple of 2. But our assumption is that \(n^{2}\) is odd, therefore we reached a contradiction, as \(4k^{2}\) is even. So, our starting assumption that \(n\) could be even was incorrect and thus \(n\) indeed must be odd when \(n^{2}\) is odd.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Even and Odd Integers
Grasping the nature of even and odd integers is essential to understand numerous mathematical concepts. Simply put, an even integer is any number that can be divided by 2 without leaving a remainder. These numbers end in 0, 2, 4, 6, or 8. For example, 2, 4, and 16 are all even numbers because they can be expressed as 2 times another integer, like so: 2=2脳1, 4=2脳2, and 16=2脳8.

On the other hand, an odd integer is a number that cannot be divided evenly by 2; it always leaves a remainder of 1. Odd numbers end in 1, 3, 5, 7, or 9. Take 3, 5, or 15, for instance. None of these can be perfectly split into two groups without having one left over. To formalize, any odd integer can be represented by 2k+1, where k is any integer鈥攖his means it's always just one more than an even number.

In the realm of proofs, understanding the clear distinction between even and odd helps in constructing logical arguments and identifying contradictions as shown in the given exercise.
Mathematical Proof
The heart of mathematics often lies in proofs, which are logical and structured arguments that establish the truth of a mathematical statement beyond any doubt. Unlike in real-world debate where evidence can leave room for interpretation, a mathematical proof provides absolute certainty.

One such proof technique is proof by contradiction. This method starts by assuming the opposite of what we want to prove and then logically showing that this assumption leads to a situation that cannot possibly be true鈥攁 contradiction. This indirectly proves that our original statement must be true.

The steps in the proof by contradiction provided in the exercise are an excellent example of leading to a contradiction from a logically incorrect assumption. By finding that an even assumption leads to an odd conclusion, it powerfully demonstrates the irreconcilable conflict within the initial assumption.
Integer Properties
Properties of integers are the fundamental rules that these numbers obey. These rules help us understand how integers behave in different mathematical scenarios. For starters, integers consist of positive and negative whole numbers, including zero. Noteworthy properties include the commutative property for addition and multiplication (rearranging order doesn't change the result), and the associative property (grouping doesn't affect the addition or multiplication of integers).

Multiple of Integers

A multiple of an integer is the result of multiplying that number by another integer. For example, 8 is a multiple of 2 because 2 multiplied by 4 is 8. Equally, this applies to even and odd properties: if an integer is multiplied by 2 (an even number), the result is always even.

The exercise used these properties to define even numbers mathematically as 2 times an integer. This definition is essential for proving whether a square of an integer is odd or even. By seeing through the lens of integer properties, the fallacy in the assumption becomes apparent when applying the definitions to the proof by contradiction.

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Most popular questions from this chapter

10\. Verify that \([p \rightarrow(q \rightarrow r)] \rightarrow[(p \rightarrow q) \rightarrow(p \rightarrow r)]\) is a tautology.

4\. For each of the following pairs of statements, use Modus 'onens or Modus Tollens to fill in the blank line so that a valid rgument is presented. a) If Janice has trouble starting her car, then her daughter Angela will check Janice's spark plugs. Janice had trouble starting her car. b) If Brady solved the first problem correctly, then the answer he obtained is 137 . Brady's answer to the first problem is not 137 . c) If this is a repeat-until loop, then the body of this loop is executed at least once. The body of the loop is executed at least once. d) If Tim plays basketball in the afternoon, then he will not watch television in the evening.

8\. Let \(p, q, r\), and \(s\) be primitive statements. Write the dual of each of the following compound statements. a) \((p \vee \neg q) \wedge(\neg r \vee s)\) b) \(p \rightarrow(q \wedge \neg r \wedge s)\) c) \(\left[\left(p \vee T_{0}\right) \wedge\left(q \vee F_{0}\right)\right] \vee\left[r \wedge s \wedge T_{0}\right]\)

1\. Let \(p, q, r\) denote primitive statements. a) Use truth tables to verify the following logical equivalences. i) \(p \rightarrow(q \wedge r) \Longleftrightarrow(p \rightarrow q) \wedge(p \rightarrow r)\) ii) \([(p \vee q) \rightarrow r] \Longleftrightarrow[(p \rightarrow r) \wedge(q \rightarrow r)]\) iii) \([p \rightarrow(q \vee r)] \Longleftrightarrow[\neg r \rightarrow(p \rightarrow q)]\) b) Use the substitution rules to show that $$ [p \rightarrow(q \vee r)] \Longleftrightarrow[(p \wedge \neg q) \rightarrow r] $$

5\. Write the following statement as an implication in two vays, each in the if-then form: Either Kaylyn practices her piano essons or she will not go to the movies.
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