Question

Write the following in set notation: (a) The set of all real numbers greater than 34 (b) The set of ail real numbers greater than 8 but less than 65

Short answer

(a) \( \{ x \in \mathbb{R} \,|\, x > 34 \} \), (b) \( \{ x \in \mathbb{R} \,|\, 8 < x < 65 \} \)

Step-by-step solution

Express the Set for Condition (a)

We need to express the set of all real numbers greater than 34. This set includes all real numbers, denoted as \( x \), such that \( x > 34 \). Therefore, this set in set notation is \( \{ x \in \mathbb{R} \,|\, x > 34 \} \).

Express the Set for Condition (b)

Next, we express the set of all real numbers greater than 8 but less than 65. This includes all real numbers, denoted as \( x \), such that \( 8 < x < 65 \). The set notation for this is \( \{ x \in \mathbb{R} \,|\, 8 < x < 65 \} \).

Key concepts

Real Numbers

Real numbers are all the numbers that make up a continuous line on the number line.
They include a variety of different types of numbers such as:

• Whole numbers (e.g., 0, 1, 2, 3, ...)
• Rational numbers, which can be expressed as fractions (e.g., 1/2, 3/4)
• Irrational numbers, which cannot be expressed as fractions (e.g., \( \pi, \sqrt{2} \))
• Negative numbers (e.g., -1, -2.5)

Real numbers are denoted by the symbol \( \mathbb{R} \). This set of numbers is crucial in mathematics because it accommodates both decimal and integer values, allowing for a comprehensive understanding of mathematical operations.
Real numbers are unending, which means there is no smallest or largest real number. They fill every possible gap on the number line, ensuring a continuous flow of values.

Inequalities

Inequalities express the relative size or order of two values. They show whether one value is greater than, less than, or possibly equal to another. Inequalities are crucial when defining ranges of numbers, such as specifying intervals on the number line.
There are several symbols used to represent inequalities:

• \( > \): greater than
• \( < \): less than
• \( \geq \): greater than or equal to
• \( \leq \): less than or equal to

For example, the inequality \( x > 34 \) means that \( x \) takes on any value that is greater than 34.
Inequality notation is especially useful in set notation where conditions can be set for values to satisfy such inequalities. This makes it easy to define a range of values foreseen by problem constraints.

Mathematical Notation

Mathematical notation is a system of symbols and signs that are used to express mathematical ideas and formulas.
It is essential in providing a clear, precise, and universally understood language in mathematics.Let's explore how it is used in set notation:

• Curly braces \( \{ \} \) are used to denote a set.
• \( \in \) symbolizes "is an element of."
• \( \mathbb{R} \) denotes the set of all real numbers.
• The vertical bar \( | \) translates to "such that," dividing the notation into conditions that must be met.

For instance, in the set notation \( \{ x \in \mathbb{R} \,|\, x > 34 \} \), we read it as 'the set of all \( x \) in real numbers such that \( x \) is greater than 34.'
Using mathematical notation helps convey complex mathematical concepts quickly and with minimum ambiguity.