Question

1. let ( p(x,y) ) be the predicate \if ( x in mathbb{r} ), then ( x geq \frac{1}{x} ).\ provide a counterexample that shows the statement is false.

1. find the truth set for the predicate: ( 60/d ) is an integer, domain ( mathbb{z}^+ ). note: ( mathbb{z}^+ ) represents the positive integers.

Explanation:

Step1: Identify counterexample condition

We need an $x \in \mathbb{R}$ where $x < \frac{1}{x}$.

Step2: Test positive fraction

Take $x = \frac{1}{2}$. Calculate $\frac{1}{x} = 2$. Since $\frac{1}{2} < 2$, this satisfies $x < \frac{1}{x}$.

Step3: List positive divisors of 60

Find all positive integers $d$ that divide 60 evenly: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

Answer:

1. A counterexample is $x = \frac{1}{2}$ (since $\frac{1}{2} < \frac{1}{\frac{1}{2}} = 2$)
2. $\{1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\}$