Question

Which of the following are true? Unless it is stated otherwise, assume that \(x, y\), and \(\varepsilon\) are real numbers. (a) For every \(x, x0\), there exists a \(y\) such that \(y>\frac{1}{x}\). (d) For every positive \(x\), there exists a natural number \(n\) such that \(\frac{1}{n}

Short answer

Statements (a), (c), (d), and (e) are true; statement (b) is false.

Step-by-step solution

Analyze Statement (a)

The statement says that for every real number \(x\), \(x < x+1\). This is true because adding 1 to any real number will always result in a number that is greater than the initial number.

Analyze Statement (b)

Statement (b) suggests that there exists a natural number \(N\) such that all prime numbers are less than \(N\). This is false as there are infinitely many prime numbers, meaning no single \(N\) can fit the condition.

Analyze Statement (c)

Statement (c) claims that for every \(x>0\), there exists a \(y\) such that \(y > \frac{1}{x}\). This is true because, given any positive real number \(x\), \(y = \frac{1}{x} + 1\) will always be greater than \(\frac{1}{x}\).

Analyze Statement (d)

The statement implies that for every positive \(x\), there exists a natural number \(n\) such that \(\frac{1}{n} < x\). This is true because for any positive \(x\), we can choose \(n\) large enough to ensure \(\frac{1}{n} < x\).

Analyze Statement (e)

Statement (e) involves every positive \(\varepsilon\), stating there exists a natural number \(n\) such that \(\frac{1}{2^{n}} < \varepsilon\). This is true, because for any positive \(\varepsilon\), we can find a sufficiently large \(n\) such that \(\frac{1}{2^{n}} < \varepsilon\).

Key concepts

Real Numbers

Real numbers are numbers that can be found on the number line. They include both rational numbers (such as integers and fractions) and irrational numbers (numbers that cannot be expressed as fractions, such as \(\sqrt{2}\) and \(\pi\)).
In essence, real numbers cover the entirety of the numbers you'll encounter in standard arithmetic and algebra. They are important because they can represent continuous quantities.
Examples include:

• Integers like -3, 0, and 5.
• Fractions like \(\frac{1}{2}\) and \(-\frac{7}{4}\).
• Irrational numbers such as \(\sqrt{3}\) and \(\pi\).

Understanding real numbers enables you to solve equations that involve a wide range of values and to comprehend continuous data, which is essential in calculus and statistics.

Natural Numbers

Natural numbers are the set of positive integers starting from 1. This means they are the numbers you typically use for counting: 1, 2, 3, and so on. In some definitions, 0 is included as a natural number.
Natural numbers are critical because:

• They are the building blocks of mathematics.
• They are used to define other number sets, like integers and rational numbers.
• They have specific properties, such as being only positive and discrete.

Natural numbers also play a pivotal role in defining prime numbers and are foundational in areas such as number theory and combinatorics.

Prime Numbers

Prime numbers are natural numbers greater than 1, which have no divisors other than 1 and themselves. This means a prime number should not be divisible by any other number.
For example, the smallest prime numbers are 2, 3, 5, and 7.
Prime numbers are essential because:

• They are the building blocks of all integers, as every integer can be factored into a product of primes.
• They have applications in cryptography, making online communications secure.
• Their unique properties are pivotal in number theory, the study of integers.

The fact that there are infinitely many prime numbers means no single natural number \(N\) can contain them all, which is a remarkable and essential concept in mathematics.

Inequalities

Inequalities describe the relative size or order of two values. In mathematics, they express relationships such as greater than (>) or less than (<).
For any real number \(x\), the statement \(x < x+1\) is a classic example of an inequality. It conveys that adding a number makes it larger than it was before.
Inequalities are important because:

• They help describe constraints and conditions in mathematical models and real-world problems.
• They are used to express intervals on the number line, such as \((a, b)\) where \(a < x < b\).
• Solving inequalities is essential in calculus when determining the range of functions.

Being comfortable with inequalities allows you to tackle problems involving systems of equations, optimization, and comparative analysis.