Question

Write each of the following sets by listing their elements between braces. $$ \left\\{x \in \mathbb{R}: x^{2}=3\right\\} $$

Short answer

The set \( \left\{x \in \mathbb{R}: x^{2}=3\right\} \) written by listing their elements between braces is \( \left\{\sqrt{3}, -\sqrt{3}\right\} \).

Step-by-step solution

Interpret the Set Definition

The set definition \( \{x \in \mathbb{R}: x^{2} = 3\} \) translates to 'the set of all real numbers \( x \) such that \( x^{2} = 3 \)'.

Solve the Given Equation for x

In this case, the equation to solve is \( x^{2} = 3 \). We can solve this equation by taking the square root of both sides. Remember that squaring a number has two solutions, positive and negative, so \( x = \sqrt{3} \) or \( x = -\sqrt{3} \).

List the Elements of the Set

The solutions for \( x \), \( \sqrt{3} \) and \( -\sqrt{3} \), are the elements of the set as they are the only real numbers that, when squared, result in 3.