Select a counterexample that makes the conclusion false.

Conclusion: The difference of two positive numbers is always positive.

Examples:
7 - 3 = 4
8 - 5 = 3
9 - 8 = 1

Counterexample: _________



Answer :

Certainly! To determine a counterexample that makes the given conclusion, "The difference of two positive numbers is always positive," false, let's go through the problem step by step:

1. Understanding the Conclusion:
The statement claims that if you subtract one positive number from another positive number, the result will always be a positive number.

2. Identifying the Task:
To disprove this conclusion, we need to find an example where the difference of two positive numbers is not positive (i.e., it is zero or negative).

3. Choosing Appropriate Numbers:
Let's select two positive numbers such that the first number is smaller than the second number. For example, let’s choose:
- The first positive number: [tex]\(3\)[/tex]
- The second positive number: [tex]\(5\)[/tex]

4. Calculating the Difference:
Subtract the first positive number from the second positive number:
[tex]\[ 3 - 5 = -2 \][/tex]

5. Analyzing the Result:
- The result of [tex]\(3 - 5\)[/tex] is [tex]\(-2\)[/tex].
- [tex]\(-2\)[/tex] is a negative number.

6. Conclusion:
Since we obtained a negative difference, this result [tex]\(3 - 5 = -2\)[/tex] serves as a counterexample that disproves the conclusion that "the difference of two positive numbers is always positive."

7. Counterexample:
Therefore, the selected counterexample is:
[tex]\[ 3, 5 \quad \text{with the difference} \quad 3 - 5 = -2 \][/tex]

By this detailed analysis, we have successfully found a counterexample that shows the conclusion is not always true.