Consider the following interpretation:
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. (∀x)(∃y)P(y, x)
c. (∀x)(∀y)(P(x, y) → P(f(x), f(y))
My attempt at solution:
a)
P(1,2) | P(2,1) | P(1,2) and P(2,1)
____________________________________
T | F | F
b)
x | y | P(y,x)
____________________
1 | 1 | T
1 | 2 | F
2 | 1 | T
2 | 2 | F
But I'm not sure if this is correct. I haven't seen anyone make a truth table yet using variables in the table. Is that doable? What else do I have to say to evaluate the truth of the entire statement?
c)
x | y | P(x,y) | P(f(x),f(y) | P(x,y) implies P(f(x), f(y)
____________________________________________________________
1 | 1 | T | F | F
1 | 2 | T | F | F
2 | 1 | F | T | T
2 | 2 | F | T | T
Again, same with b), I don't know if I'm taking the right approach.
Thanks!