Question

Write out the indicated sets by listing their elements between braces. $$ \left\\{x \in \mathbb{R}: x^{2}=2\right\\} \times\\{x \in \mathbb{R}:|x|=2\\} $$

Short answer

The Cartesian product of the two sets is \(\{(-\sqrt{2}, -2), (-\sqrt{2}, 2), (\sqrt{2}, -2), (\sqrt{2}, 2)\}\).

Step-by-step solution

Solve the first set

Solving \(x^{2} = 2\) will give two solutions, \(x = \sqrt{2}\) and \(x = -\sqrt{2}\). So the first set is \(\{-\sqrt{2}, \sqrt{2}\}.\)

Solve the second set

Solving \(|x| = 2\) also gives two solutions. For \(x\) to have an absolute value of 2, \(x\) could be either 2 or -2. So the second set is \(\{-2, 2\}\).

Compute the Cartesian product

Now multiply the two sets together using the Cartesian product. This results in the following pairs: (-\(\sqrt{2}\),-2), (-\(\sqrt{2}\),2), (\(\sqrt{2}\),-2), (\(\sqrt{2}\),2). So the Cartesian product of the two sets is \(\{(-\sqrt{2}, -2), (-\sqrt{2}, 2), (\sqrt{2}, -2), (\sqrt{2}, 2)\}\).

Key concepts

Set Theory

Set theory is a fundamental part of mathematics that deals with collections of objects, known as sets. A set can be defined explicitly by listing its elements or described implicitly by the properties that its members must satisfy.

For instance, a set containing the numbers 1, 2, and 3 would be written as \(\{1, 2, 3\}\). In our exercise, sets are described by the properties their members must satisfy, where one property is related to the squares of real numbers (\(x^{2} = 2\)) and another to the absolute value (\(|x| = 2\)).

The exercise also introduces the concept of the Cartesian product. This operation, represented by the symbol \(\times\), combines two sets to form a new set of ordered pairs. Each pair contains one element from each set, with the first set providing the first element of each pair, and the second set providing the second element. Think of it as mixing two decks of cards, where you create a pair by picking one card from each deck.

Absolute Value

The absolute value represents the distance of a number from zero on the real number line, regardless of direction. It’s always non-negative.

For example, the absolute value of both -2 and 2 is 2, denoted as \(|-2| = 2\) and \(|2| = 2\). In our exercise, solving the equation \(|x| = 2\) means finding all real numbers that are two units away from zero, which yields \(x=2\) and \(x=-2\), forming a set \(\{-2, 2\}\) as the solution. Understanding absolute value is vital because it helps explain why certain equations, especially those involving absolute values, have more than one solution.

Real Numbers

Real numbers encompass all the numbers on the number line, including both rational numbers (like \(\frac{1}{2}\) or 3) and irrational numbers (such as \(\sqrt{2}\) or \(\pi\)). The set of real numbers is denoted by \(\mathbb{R}\).

In the exercise, the notation \(x \in \mathbb{R}\) indicates that the elements we are looking for are real numbers. This is an important distinction because the way we solve equations can depend on the number set we are restricted to. If we were restricted to rational numbers, for example, we wouldn’t be able to include \(\sqrt{2}\) since it's an irrational number.

Square Root

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, since \(2 \times 2 = 4\), we say that 2 is a square root of 4. Every positive real number has two square roots, one positive and one negative, since both \((\sqrt{x})^{2}\) and \((-\sqrt{x})^{2}\) equal \(x\).

In our problem, we are looking for all square roots of 2. This means solving \(x^{2} = 2\), and it leads us to two solutions: the positive square root \(\sqrt{2}\), and the negative square root \(-\sqrt{2}\). These solutions are then used to create a set that is part of the Cartesian product in the exercise.