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.
