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Chapter 4: Problem 11

Prove that there exists no integer between 0 and \(1 .\)

Short Answer

Expert verified
Using proof by contradiction, we assume that there exists an integer \(x\) in the range \(0 < x < 1\). However, this contradicts the definition of integers since \(x\) should either be less than or equal to 0 or greater than or equal to 1. As a result, our assumption is false, and we conclude that there is no integer between 0 and 1.

Step by step solution

01

Understanding Integers

Integers are whole numbers, and they can be either positive, negative, or zero. The set of integers is represented by the symbol \(\mathbb{Z}\) and can be written as: \(\mathbb{Z} = \{...,-3, -2, -1, 0, 1, 2, 3, ...\}\). Note that integers do not include any fractional or decimal values.
02

Between 0 and 1

Now let's consider the given problem, which states that there exists no integer between 0 and 1. We need to demonstrate that there are no whole numbers in this range.
03

Proof by Contradiction

To prove that there are no integers in the range (0, 1), let's use a proof by contradiction. Assume that there does exist an integer between 0 and 1. Let this integer be x, such that \(0 < x < 1\).
04

Contradiction

Since x is an integer, it should either be less than or equal to 0 or greater than or equal to 1. But according to our assumption, it falls in the range (0, 1), meaning it is greater than 0 and less than 1. This contradicts the definition of an integer.
05

Conclusion

Therefore, we have a contradiction, and our assumption that an integer exists between 0 and 1 must be false. Hence, we have proven that there is no integer between 0 and 1.

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Key Concepts

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

Proof by Contradiction
Proof by contradiction is a fascinating way to prove a statement is true. It works by assuming the opposite of what you're trying to prove, and then showing that this assumption leads to a contradiction.
In simpler terms, if assuming something leads to an impossible situation or a false conclusion, then the original statement must be true. It's like solving a puzzle backwards.

For example, in our problem, we started by assuming there is an integer between 0 and 1. Through logical reasoning, we found that no such integer can exist, because integers are supposed to be whole numbers and must be equal to or beyond 0 and 1. The assumption leads us to a conclusion that contradicts what we know about integers. By proving that this assumption leads to an absurdity, we confirmed that the original statement is indeed correct.
Set of Integers
The set of integers is a collection of numbers that can be positive, negative, or zero. These are numbers without fractional or decimal parts.
Integers form a complete set and are denoted by the symbol \( \mathbb{Z} \). When we write them out, they look like:
  • ..., -3, -2, -1, 0, 1, 2, 3, ...

This means integers go on infinitely in both the positive and negative directions. However, they never include numbers like 0.5, 1.2, or any fractional numbers.
Understanding the nature of integers is crucial for solving problems about continuous number ranges. This particularity of integers is crucial in our problem since no integers can exist between any two consecutive whole numbers, such as 0 and 1.
Whole Numbers
Whole numbers are a specific category of numbers that include all positive numbers and zero, but they don't include negative numbers.
They start from 0 and move upwards like this:
  • 0, 1, 2, 3,...

Whole numbers are simple to understand because they are the numbers we often use in counting.
In our example, since the problem is about finding an integer between 0 and 1, knowing whole numbers helps in understanding that there can't be a whole number like that.
This is because whole numbers jump directly from 0 to 1, with no possible integers in between, hence supporting the outcome of our proof by contradiction.

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Most popular questions from this chapter

Using the euclidean algorithm, find the gcd of the given integers. $$3076,1976$$

Let \(n_{0} \in \mathbf{Z}, S\) be a nonempty subset of the set \(T=\left\\{n \in \mathbf{Z} | n \geq n_{0}\right\\}\) and \(\ell^{*}\) be a least element of the set \(T^{*}=\left\\{n-n_{0}+1 | n \in T\right\\} .\) Prove that \(n_{0}+\ell^{*}-1\) is a least element of \(S\)

Evaluate each sum, where \(d\) is a positive integer. $$\sum_{d | 12} 1$$

The binary representation of an integer can conveniently be used to find its octal representation. Group the bits in threes from right to left and replace each group with the corresponding octal digit. For example, $$243=11110011_{\text {two }}=011 \quad 110 \quad 011_{\text {two }}=363_{\text {eight }}$$'Using this short cut, rewrite each binary number as an octal integer. $$1101_{\text {two }}$$

Euler's phi-function \(\varphi\) is another important number-theoretic function on \(\mathbb{N},\) defined by \(\varphi(n)=\) number of positive integers \(\leq n\) and relatively prime to \(n .\) For example, \(\varphi(1)=1=\varphi(\mathbf{2}), \varphi(3)=\mathbf{2}=\varphi(4),\) and \(\varphi(5)=4 .\) Evaluate \(\varphi(n)\) for each value of \(n\). $$15$$
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