Question

PREDICATE LOGIC.(25 pts)

A. Let P(x) be the statement x

2 > x4. If the domain consists of the integers, 

what are the truth values? 

1. P(0) 

2. P(-1) 

3. P(1) 

4. P(2) 

5. ∃xP(x)

6. ∀xP(x)

Answer 1

Let $P(x)$ be the statement $x^2 > x^4$. If the domain consists of the integers, let us find the truth values:

1. Since it is not true that $0>0$, we conclude that $P(0) =F.$

2. Since $(-1)^2 =1= (-1)^4$, we conclude that $P(-1) =F.$

3. Since $1^2 =1= 1^4$, we conclude that $P(1) =F.$

4. Since $2^2=4<16=2^4$, we conclude that $P(2) =F.$

5. Since $x^2\le x^4$ for any integer $x$, we conclude that $∃xP(x)=F.$

6. Since $x^2\le x^4$ for any integer $x$, we conclude that $∀xP(x)=F.$