Evaluate truth value of formula

Question

I am not sure about this question?

Domain = {1, 2}

Assignment of constants: a = 1 and b = 2

Assignment of functions: f(1) = 2 and f(2) = 1

Assignment for predicate P: P(1, 1) = T; P(1, 2) = T; P(2, 1) = F; P(2, 2) = F

Evaluate the truth value of following formulas in the above interpretation:

a. P(a, f(a)) ∧ P(b, f(b))

b. (∀a)(∃b)P(b, a)

c. (∀a)(∀b)(P(a, b) → P(f(a), f(b))

Answer 1

Let's see:

a) $$P(a, f(a)) \land P(b, f(b)) \equiv P(1,2) \land P(2,1) \equiv T \land F \equiv F$$

b) $\forall a \exists b\; P(b,a)$?

It is true. Why? Because, for all $a$, exists $b$ such as $P(b,a)$ is true:

• If $a = 1$, then $b = 1$ verifies $P(b,a)$.

• If $a = 2$, then $b = 1$ verifies $P(b,a)$.

c) $\forall a\;\forall b \;\Big(P(a,b) \to P\big(f(a), f(b)\big)\Big)$.

Take $a = 1$ and $b = 1$ and you will see that the statement is false (counterexample): $$P(a,b) \equiv T \textbf{ does not implies } P\big(f(a), f(b)\big) \equiv P\big(f(1),f(1)\big) \equiv P(2,2) \equiv F$$