/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 4 Prove that for every \(n \in \ma... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 4

Prove that for every \(n \in \mathbf{Z}\), if \(n^{2}\) is odd, then \(n\) is odd.

Short Answer

Expert verified
The statement 'if \(n^{2}\) is odd, then \(n\) is odd' for every integer \(n\) is true and this is proven by its contraposition 'if \(n\) is even, then \(n^{2}\) is even'.

Step by step solution

01

Assumption

Assume that \(n\) is even. This means \(n\) can be written in the form \(n = 2k\), where \(k\) is an integer since all even numbers can be written as 2 times some integer.
02

Calculating Square

Square the number \(n\), we get \(n^{2} = (2k)^{2} = 4k^{2}\). This is another integer (say \(m\)) multiplied by 2, hence it is also an even number. Consequently, if \(n\) is even then \(n^{2}\) is also even.
03

Conclusion

Thus, our contrapositive statement 'if \(n\) is not odd (meaning \(n\) is even), then \(n^{2}\) is not odd (meaning \(n^{2}\) is even)' is proved. Therefore, the original statement 'if \(n^{2}\) is odd then \(n\) is odd' is also proved by the principle of contraposition.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Integers
In mathematics, integers are a fundamental concept. They are whole numbers that can be positive, negative, or zero.
Some examples include -3, 0, and 5. Integers do not include fractions or decimals. Understanding integers is essential in number theory, as they form the building blocks for various mathematical operations and proofs.
  • The set of integers is often denoted by the symbol \(\mathbf{Z}\). This comes from the German word "Zahlen," meaning numbers.
  • Integers are a subset of the real numbers, specifically containing all the natural numbers, their negative counterparts, and zero.
Working with integers allows mathematicians to construct equations, explore number properties, and develop proofs across different branches of mathematics.
Odd and Even Numbers
Understanding the distinction between odd and even numbers is crucial to number theory. An even number is divisible by 2 with no remainder, while an odd number has a remainder of 1 when divided by 2.
  • Even numbers can be expressed in the form \(n = 2k\), where \(k\) is an integer.
  • Odd numbers are expressed as \(n = 2k + 1\).
This characterization is useful for proving various mathematical statements. When squaring an odd number, the result remains odd. Conversely, squaring an even number always results in an even number.
These properties help in constructing proofs like the one discussed in our exercise.
Proof by Contraposition
Proof by contraposition is a powerful technique in logic and mathematics. It is used to prove conditional statements of the form "if \(P\), then \(Q\)." Instead of proving this directly, one proves the contrapositive: "if not \(Q\), then not \(P\)."
  • This method is valid because a statement is logically equivalent to its contrapositive.
  • It can be especially helpful when the contrapositive is easier to establish than the original statement.
In our given exercise, proving that if \(n\) is not odd (meaning even), then \(n^2\) is even, allows us to conclude that if \(n^2\) is odd, then \(n\) must be odd. This method simplifies the proof process significantly.
Mathematical Proofs
Mathematical proofs are logical arguments that establish the truth of a mathematical statement. They are a cornerstone of mathematical rigor and are used to verify theories, conjectures, and statements within the realm of mathematics.
  • Proofs can be constructed using various methods, such as direct proof, proof by contradiction, and proof by contraposition.
  • Each type of proof has its own strategic advantages, depending on the context of the problem.
A proof must begin with known facts or assumptions and use logical reasoning to arrive at a conclusion. In our exercise, a proof by contraposition effectively demonstrates the relationship between the squaring of integers and the nature of odd and even numbers. Proofs are essential for confirming that mathematical statements hold true under all conditions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Prove that the following are equivalent for sets \(A, B,\) and \(C\) : $$ \text { (a) } A \cup B=U $$ (b) \(\bar{A} \cap \bar{B}=\varnothing\) (c) \(\bar{A} \subset B\), where \(U\) is a universal set.

Prove that the product of two consecutive integers is even.

Use induction to prove that if \(X_{1}, \ldots, X_{n}\) and \(X\) are sets, then (a) \(X \cap\left(X_{1} \cup X_{2} \cup \cdots \cup X_{n}\right)=\left(X \cap X_{1}\right) \cup\left(X \cap X_{2}\right) \cup \cdots \cup\left(X \cap X_{n}\right)\) (b) \(\overline{X_{1} \cap X_{2} \cap \cdots \cap X_{n}}=\overline{X_{1}} \cup \overline{X_{2}} \cup \cdots \cup \overline{X_{n}}\)

Prove that \(2 m+5 n^{2}=20\) has no solution in positive integers.

Prove that for every \(n \in Z, n^{3}+n\) is even.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.