Question

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

Short answer

The elements of the set \(\{\sqrt{2},-\sqrt{2}\}\times\{a,c,e\}\) are \(\{(\sqrt{2},a),(\sqrt{2},c),(\sqrt{2},e),(-\sqrt{2},a),(-\sqrt{2},c),(-\sqrt{2},e)\}\).

Step-by-step solution

Find the solutions for the equation

The equation \(x^{2} = 2\) will render two solutions as \(x = \sqrt{2}\) and \(x = -\sqrt{2}\). Thus the set \(\{x \in \mathbb{R}: x^{2}=2\}\) becomes \(\{ \sqrt{2}, -\sqrt{2}\}\).

Understand the Cartesian product

Cartesian product of two sets A and B (denoted as A x B) is the set of all ordered pairs (a, b) where a is in A and b is in B. Therefore, to find \(\{\sqrt{2},-\sqrt{2}\}\times\{a,c,e\}\), we need to make pairs where the first element is from the set \(\{\sqrt{2},-\sqrt{2}\}\) and the second element is from the set \{a, c, e\}.

List the elements of the Cartesian product

The Cartesian product is \(\{(\sqrt{2},a),(\sqrt{2},c),(\sqrt{2},e),(-\sqrt{2},a),(-\sqrt{2},c),(-\sqrt{2},e)\}\).