Chapter 1, Problem 27SE

Express each of these statements using existential and universal quantifiers and propositional logic, where ${\exists }_n$ is defined in Exercise 26.

a) ${\exists }_{0}xP\left(x\right)$

b) ${\exists }_{1}xP\left(x\right)$

c) ${\exists }_{2}xP\left(x\right)$

d) ${\exists }_{3}xP\left(x\right)$

26. The quantifier ${\exists }_n$ denotes "there exists exactlyn," so that

${\exists }_{n}xP\left(x\right)$ means there exist exactlynvalues in the domain such thatP(x) is true. Determine the true value of these statements where the domain consists of all real numbers.

a) ${\exists }_{0}x\left(x^2=-1\right)$

b) ${\exists }_{1}x\left(\left|x\right|=0\right)$

c) ${\exists }_{2}x\left(x^2=2\right)$

d)

   ${\exists }_{3}x\left(x=\left|x\right|\right)$