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Chapter 1: Problem 9

Use a proof by contradiction to prove that the sum of an irrational number and a rational number is irrational.

Short Answer

Expert verified
The sum of an irrational number and a rational number is irrational.

Step by step solution

01

State the Claim

We want to prove that the sum of an irrational number and a rational number is irrational.
02

Assume the Opposite

Assume that the sum of an irrational number and a rational number is rational. We will call this sum and aim to find a contradiction.
03

Define Variables

Let the irrational number be denoted as \( x \) and the rational number as \( y \). Assume their sum is rational, say \( z \). Hence, we write: \[ x + y = z \]
04

Rearrange the Equation

Solve for the irrational number \( x \): \[ x = z - y \].
05

Analyze the Result

Since both \( z \) and \( y \) are rational numbers, their difference \( z - y \) must also be a rational number.
06

Reach a Contradiction

Our assumption leads to the conclusion that \( x \) is rational, which contradicts the original premise that \( x \) is irrational.
07

Conclude the Proof

Since assuming that the sum of an irrational number and a rational number is rational leads to a contradiction, it must be that the sum of an irrational number and a rational number is irrational. Therefore, the original claim is proven.

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Key Concepts

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

irrational numbers
Irrational numbers are numbers that cannot be expressed as a fraction of two integers. This means that they cannot be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \).
They have non-repeating, non-terminating decimals. For example, the number \( \pi \) or the square root of 2 (\( \sqrt{2} \)) are both irrational.
  • They can't be written as simple fractions.
  • The decimal goes on forever without repeating.
To better understand this, you could think of \( \pi = 3.141592653589... \). There鈥檚 no repeating pattern, and it never ends.
rational numbers
Rational numbers are the opposite of irrational numbers. They can be expressed as a fraction of two integers, \( \frac{a}{b} \), where both \( a \) and \( b \) are integers and \( b eq 0 \) . Some examples include \( \frac{1}{2} \), \( 4 \), and \( -7 \), all of which can be written as fractions.
Typical rational numbers include:
  • Whole numbers: 0, 1, 2, 3...
  • Fractions: \( \frac{3}{4} \), \( \frac{2}{5} \)
  • Terminating decimals: 0.5, 3.75
  • Repeating decimals: 0.333鈥�, 1.272727...
A key property of rational numbers is that they are either terminating or repeating decimals. This makes them easier to deal with in many cases because they have a predictable pattern.
contradiction
Contradiction is a method used in proofs to demonstrate that a statement must be true because the assumption that it is false leads to an illogical conclusion. Here's how it works in steps:
  • Assume the opposite of what you want to prove.
  • Show that this assumption leads to a contradiction or an impossibility.
  • Conclude that the original statement must be true since the opposite is false.
For example, in the provided exercise, we assumed that the sum of an irrational number and a rational number was rational. We then showed that this assumption led to a contradiction since it implied that an irrational number was rational. Because this is impossible, we concluded that the sum of an irrational number and a rational number must be irrational.

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Most popular questions from this chapter

Exercises \(40-44\) deal with the translation between system specification and logical expressions involving quantifiers. Express each of these system specifications using predicates, quantifiers, and logical connectives. a) Every user has access to an electronic mailbox. b) The system mailbox can be accessed by everyone in the group if the file system is locked. c) The firewall is in a diagnostic state only if the proxy server is in a diagnostic state. d) At least one router is functioning normally if the throughput is between 100 kbps and 500 kbps and the proxy server is not in diagnostic mode.

Show that the two statements \(\neg \exists x \forall y P(x, y)\) and \(\forall x \exists y \neg P(x, y),\) where both quantifiers over the first variable in \(P(x, y)\) have the same domain, and both quantifiers over the second variable in \(P(x, y)\) have the same domain, are logically equivalent.

Prove that given a real number \(x\) there exist unique numbers \(n\) and \(\epsilon\) such that \(x=n-\epsilon, n\) is an integer, and \(0 \leq \epsilon<1 .\)

Exercises \(61-64\) are based on questions found in the book Symbolic Logic by Lewis Carroll. Let P(x), Q(x), R(x), and S(x) be the statements 鈥渪 is a baby,鈥� 鈥渪 is logical,鈥� 鈥渪 is able to manage a crocodile,鈥� and 鈥渪 is despised,鈥� respectively. Suppose that the domain consists of all people. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), R(x), and S(x). a) Babies are illogical. b) Nobody is despised who can manage a crocodile. c) Illogical persons are despised. d) Babies cannot manage crocodiles. e) Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?

Determine whether each of these arguments is valid. If an argument is correct, what rule of inference is being used? If it is not, what logical error occurs? a) If \(n\) is a real number such that \(n>1,\) then \(n^{2}>1\) Suppose that \(n^{2}>1 .\) Then \(n>1\) b) If \(n\) is a real number with \(n>3,\) then \(n^{2}>9\) . Suppose that \(n^{2} \leq 9 .\) Then \(n \leq 3\) . c) If \(n\) is a real number with \(n>2,\) then \(n^{2}>4\) . Suppose that \(n \leq 2 .\) Then \(n^{2} \leq 4 .\)
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