Question

Write the following as English sentences. Say whether they are true or false. $$ \forall x \in \mathbb{R}, \exists n \in \mathbb{N}, x^{n} \geq 0 $$

Short answer

The translated statement is: 'For every real number, there exists a natural number such that the real number raised to the power of the natural number is greater than or equal to zero.' This statement is true.

Step-by-step solution

Translation

Firstly, translate the mathematical statement into English. The statement can be translated as: 'For all x that is a real number, there exists a natural number n such that x raised to the power of n is greater than or equal to zero.'

Analyze and Determine Truth Value

Analyze the translated statement to ascertain whether it is true or false. We know that real numbers include both positive and negative numbers - and zero. When negative numbers are raised to an even power, they result in a positive number. When they are raised to an odd power, they remain negative. Zero, raised to any power, remains zero. Therefore, for any negative real number, we can always find an even natural number which, when the negative number is raised to the power of, results in a non-negative number. Therefore, the statement is true.

Key concepts

Real Numbers

The set of real numbers, denoted by \( \mathbb{R} \), contains all the numbers that can be found on the number line. This includes all the integers, fractions, irrational numbers like \( \pi \) and \(e\), and even radicals like \( \sqrt{2} \). Real numbers are foundational to various mathematical concepts because they form a complete and ordered field.

In relation to the exercise, when we talk about real numbers, we're including every possible decimal number, whether it’s positive, negative, or zero. A crucial point for students to understand is that regardless of how small or large, or whether it's a whole number or a decimal, if it can be placed on the number line, it's a real number. When we perform operations with real numbers, the results follow certain rules and properties, which leads into our discussion about exponentiation, a common operation involving real numbers.

Natural Numbers

Natural numbers, symbolized by \( \mathbb{N} \), are the simplest set of numbers consisting of all positive integers from 1 moving upwards without end (1, 2, 3, ...). They're sometimes called 'counting numbers' because we use them naturally when we count objects. It's important to note, the set of natural numbers does not include zero or any negative numbers.

The given exercise considers whether powers of real numbers with natural number exponents can be negative. The inclusion of natural numbers is significant because they do not accommodate negative exponents or fractions - which could yield different results when combined with real numbers in exponentiation.

Exponentiation Properties

Exponentiation is a mathematical operation involving two numbers, where one number is raised to the power of a second number. For example, in \(x^n\), \(x\) is the base, and \(n\) is the exponent. If the base is a real number and the exponent is a natural number, certain properties guarantee the result will behave predictably.

One hallmark property is that any real number raised to an even exponent will result in a non-negative number. This is why the translated statement from the exercise is true: a negative base raised to an even natural number will yield a positive result, and a positive base, or zero, will always yield a non-negative result when raised to any natural number exponent. Understanding this property is fundamental for students to recognize patterns and solve more complex problems involving exponentiation.