91Ó°ÊÓ

Is the statement "There is an integer greater than 2 such that \((x-2)^{2}+1 \leq 2\) " true or false? How do you know?

Write the statement "The square of every real number is greater than or equal to \(0 "\) " as a quantified statement about the universe of real numbers. You may use \(R\) to stand for the universe of real numbers.

A prime number is defined as an integer greater than 1 whose only positive integer factors are itself and 1. Find two ways to write this definition so that all quantifiers are explicit. (It may be convenient to introduce a variable to stand for the number and perhaps a variable or some variables for its factors.)

Prove that \(\sqrt{3}\) is irrational.

Let \(p(x)\) stand for " \(x\) is a prime," \(q(x)\) for " \(x\) is even," and \(r(x, y)\) stand for " \(x=y . "\) Use these three symbolic statements and appropriate logical notation to write the statement "There is one and only one even prime." (Use the set \(Z^{+}\)of positive integers for your universe.)