Answer
a. "There are real numbers whose sum is less than their difference." This statement is true: for example, $3 + (-5) \lt 3 - (-5)$, because $-2 \lt 8$.
b. "There is a real number whose square is less than the original number." This statement is true: for example, $(\frac{1}{3})^{2} \lt \frac{1}{3}$, because $\frac{1}{9} \lt \frac{1}{3}$.
c. "The square of any positive integer is greater than or equal to that integer." This statement is true, and the only positive integer equal to its square is $1$. For all other positive integers, the strict inequality holds.
d. "The absolute value of the sum of any two real numbers is less than or equal to the sum of their absolute values." This statement, known as the triangle inequality, is true.
Work Step by Step
The goal of this exercise is to phrase the given statements so that they are "less intimidating," so to speak. Although this is subjective, the general idea is to eliminate mathematical variables and jargon while maintaining the precision of the original statement.
