Question

question 5
to prove that a conjecture is false, you need to provide a counterexample. a counterexample is an example that shows a conjecture is false. which of the following conjectures is true?
a) for all real numbers ( x ), the expression ( x^2 ) is greater than or equal to ( x ).
b) all prime numbers are odd.
c) if the product of two numbers is positive, then the two numbers must be both positive.
d) the product of two negative numbers is always positive.
e) if the product of two numbers is even, then the two numbers must be even.
f) none of the above.

Explanation:

• For option a: Let \( x = 0.5 \), then \( x^{2}=0.25 \), and \( 0.25<0.5 \), so the conjecture is false.
• For option b: The prime number 2 is even, so the conjecture "All prime numbers are odd" is false.
• For option c: The product of two negative numbers is positive (e.g., \((-2)\times(-3) = 6\)), so the conjecture "If the product of two numbers is positive, then the two numbers must be both positive" is false (since two negative numbers also give a positive product).
• For option d: By the rule of multiplying negative numbers (\((-a)\times(-b)=ab\) where \(a,b>0\)), the product of two negative numbers is always positive. Also, if we consider a negative and a positive number, their product is negative, and the product of two positive numbers is positive. So for any two negative numbers, their product is positive, and there is no counterexample to disprove this. So this conjecture is true.
• For option e: Consider \( 3\times5 = 15 \) (product is odd, but both numbers are odd), so the conjecture "If the product of two numbers is even, then the two numbers must be even" is false.

Answer:

d) The product of two negative numbers is always positive.