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. Taking into account that it is not true that $0>0$, we conclude that $P(0) =F.$

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

3. Taking into account that $1^2 =1= 1^4$, we have that $P(1) =F.$

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

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

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