Final answer:
The statements a) and b) are false as √2 is irrational, c is false because 15 is not a prime number, d is true as it correctly implies 15 is not prime, e) is true for non-positive real numbers, and f is logically true but factually unrelated.
Step-by-step explanation:
Let's go through each statement and determine if it's true or false, providing justification for the answers:
a) False. √2 is irrational regardless of whether 2 is a perfect square or not. In fact, since 2 is not a perfect square, its square root cannot be expressed as a ratio of two integers. √2 is a well-known example of an irrational number.
b) False. If 2 were a perfect square, then √2 would be rational because it would be the square root of a number that can be expressed as a whole number. However, 2 is not a perfect square, and thus √2 is irrational.
c) False. The multiplication of 5 and 3 being equal to 15 is not necessary for 15 to be a prime number. In fact, this multiplication shows that 15 has divisors other than 1 and itself, so it is not prime.
d) True. If 5 times 3 is not equal to 15, then 15 must have some other factors, which means it cannot possibly be a prime number. Thus, this condition is sufficient for 15 to not be prime.
e) True. The statement |x| = -x if and only if √x² =-x is true when x is a non-positive real number (x ≤ 0). For non-positive real numbers, the absolute value of x is indeed -x, and the square root of x squared is also the negative of x, due to the non-positive assumption.