Problem 60 - Algebra 2

Question

Determine whether each statement is true or false. If false, give a counterexample. A counterexample is a specific case that shows that a statement is false. Every integer is a whole number.

Step-by-Step Solution

Answer

False. Counterexample: -1 is an integer but not a whole number.

Step 1: Identify Definitions

First, let's establish what integers and whole numbers are. Integers include all positive and negative whole numbers, including zero, i.e., \( ightarrow \{-3, -2, -1, 0, 1, 2, 3, \} \). Whole numbers, on the other hand, are all the non-negative integers, i.e., \( ightarrow \{0, 1, 2, 3, \} \).

Step 2: Analyze the Statement

The statement claims that every integer is also a whole number. Based on the definitions, let's analyze if all integers fit the definition of whole numbers.

Step 3: Check for Counterexamples

Now, check for integers that do not fit the definition of whole numbers. Since whole numbers are non-negative, any negative integer will act as a counterexample. For example, the integer \(-1\) is negative and therefore not a whole number.

Step 4: Evaluate and Conclude

Since we found that negative integers, such as \(-1\), are not whole numbers, we conclude the statement is false. Having identified a counterexample, the assertion does not hold universally.

Key Concepts

Whole Numbers

Whole numbers are like the starting lineup for the number team! They include:

• 0
• Positive numbers like 1, 2, 3, and so on

Whole numbers do not have fractions, decimals, or negatives. Imagine them as counting numbers—anything you can count on your fingers. These are the numbers you start learning about when you first learn math! Whole numbers begin at zero and go on forever, but they never dip below zero.

Counterexamples

A counterexample is like a detective clue that a statement isn't true. When someone says, "All integers are whole numbers," you look for one piece of evidence to prove them wrong.

Consider negative integers, like -1. Since whole numbers don’t include negative numbers, finding one negative integer proves the statement false.

• Counterexamples answer the true or false question.
• All it takes is one to disprove a claim.

Finding a counterexample is a powerful tool in logic and math because it pinpoints exactly why an assumption doesn’t hold up under scrutiny.

Mathematical Definitions

Mathematical definitions are the rules that tell us what something means in math. They're like the guidelines or instructions.

For integers and whole numbers:

• Integers: All positive and negative whole numbers, including zero.
• Whole Numbers: All non-negative integers starting from zero upwards.

Having clear definitions helps to avoid confusion. This is why, when solving any math-related questions, knowing these terms is incredibly important. It guides you to understand what’s included and what’s not in a set or category.

Negative Integers

Negative integers might sound gloomy, but they play an important role in math! These are simply whole numbers with a minus sign in front of them, like -1, -2, or -3.

• They're used to represent values less than zero—think of temperatures below freezing or debts.
• Negative integers help expand arithmetic beyond just counting and measuring.

Importantly, while integers include negative numbers, whole numbers don’t. That's why a negative integer like -3 shows that not all integers are whole numbers.