Answered

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3. Which counterexample shows the conjecture "If the product of two numbers is positive, then the two numbers must both be positive" to be false?

A. [tex]\((-3)(+1) = -3\)[/tex]
B. [tex]\((-3)(-1) = +3\)[/tex]
C. [tex]\((+3)(+1) = +8\)[/tex]
D. [tex]\((+3)(-1) = -3\)[/tex]

Sagot :

GinnyAnswer
To determine which counterexample shows the conjecture "If the product of two numbers is positive, then the two numbers must both be positive" to be false, let's examine each product presented:

1. [tex]\( (-3)(+1) = -3 \)[/tex]
- The product is negative. This does not contradict the conjecture since the product is not positive.

2. [tex]\( (-3)(-1) = +3 \)[/tex]
- The product is positive, but the two numbers are not both positive (both numbers are negative). This counterexample contradicts the conjecture because it shows that a product can be positive even when the numbers are not both positive.

3. [tex]\( (+3)(+1) = +3 \)[/tex]
- The product is positive, and both numbers are positive. This does not contradict the conjecture.

4. [tex]\( (+3)(-1) = -3 \)[/tex]
- The product is negative. This does not contradict the conjecture since the product is not positive.

Thus, the counterexample that shows the conjecture to be false is:

[tex]\[ (-3)(-1) = +3 \][/tex]

This example demonstrates that two numbers can multiply to produce a positive product even when both numbers are not positive.
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