Problem 37 - Intermediate Algebra

Question

List the elements of each set. For example, the elements of \(\\{x \mid x\) is a natural number less than 4\(\\}\) can be listed as \(\\{1,2,3\\}\). \(\\{y \mid y\) is an integer less than 3\(\\}\)

Step-by-Step Solution

Answer

\( \{ ..., -3, -2, -1, 0, 1, 2 \} \)

Step 1: Understand the Set Notation

This set is defined by a condition: \( \{ y \mid y \text{ is an integer less than 3} \} \). This means we are looking for all integer values of \( y \) that are less than 3.

Step 2: Identify Integers

Integers are the whole numbers including negative numbers, zero, and positive numbers (e.g., ..., -2, -1, 0, 1, 2, ...). We need to determine which of these integers are less than 3.

Step 3: List Integers Less Than 3

Start from 3 and move downwards listing integers: 2, 1, 0, -1, -2, -3, ... These are all the integers smaller than 3.

Step 4: Express the Set

The set of integers less than 3 can be written as \( \{ ..., -3, -2, -1, 0, 1, 2 \} \). These are all the integer values of \( y \) that satisfy the condition \( y < 3 \).

Key Concepts

Integer

Integers are one of the most basic concepts in mathematics. They include all whole numbers and their negative counterparts. In mathematical terms, integers are a set consisting of:

• Positive whole numbers
• Negative whole numbers
• Zero

For example, \[ ext{Integers} = \{... -3, -2, -1, 0, 1, 2, 3, ...} \]
This is very different from fractions or decimals. When we say numbers like -5, 0, and 10, we are referring to integers. Understanding this set is crucial because integers form the basis for more complex number systems and are fundamental in mathematical operations like addition, subtraction, and much more. Learning about integers allows you to appreciate this infinite line of numbers that stretches both to the left and to the right of zero.

Set Notation

Set notation is a way of describing a collection of objects. This is very handy when you want to communicate precisely which elements belong to a group. Sets can be expressed with curly braces, such as \( \{1, 2, 3\} \), indicating a set containing 1, 2, and 3.
There are several ways to describe sets. One such way is by listing all elements, known as the roster method. For example, \( \{x \, | \, x \, \text{is a natural number less than 4} \} = \{1, 2, 3\} \). This tells us just the elements you can find in this particular group.
Alternatively, you can describe a set by a property that its members must satisfy. For instance, \( \{y \mid y \, \text{is an integer less than 3} \} \) means y could be any integer that is smaller than 3. This notation helps in simplifying the representation and gives you a flexible way of handling different kinds of mathematical conditions.

Mathematical Logic

Mathematical logic forms the foundation of mathematical reasoning and proofs. It's about using structured principles to derive truths and solve problems. One of the core ideas in logic is conditional statements, which you saw in the original exercise with statements like "y is an integer less than 3."
Mathematical logic includes operations like:

• Conjunctions ("AND")
• Disjunctions ("OR")
• Negations ("NOT")

In set theory, logic helps dictate membership criteria, determining which elements belong to a particular set. Through logical expressions or set-builder notation, as in \( \{y \mid y < 3\} \), you specify exactly what qualifies an element as a part of a set.
Logic ensures clarity and precision in mathematics, making it easier for you to understand complex ideas and develop arguments grounded in evidence and well-defined rules. These logical principles are widely used beyond math, such as in computer science, philosophy, and linguistics.