The tables below show the values of predicates P(x, y), Q(x, y), and S(x, y) for every possible combination of values of the variables x and y. The row number indicates the value for x and the column number indicates the value for y. The domain for x and y is {1, 2, 3}.
P 1 2 3
1 T F T
2 T F T
3 T T F
Q 1 2 3
1 F F F
2 T T T
3 T F F
S 1 2 3
1 F F F
2 F F F
3 F F F
Indicate whether each of the quantified statements is true or false.
(a)
∃x ∀y P(x, y)
(b)
∃x ∀y Q(x, y)
(c)
∃x ∀y P(y, x)
(d)
∃x ∃y S(x, y)
(e)
∀x ∃y Q(x, y)
(f)
∀x ∃y P(x, y)
(g)
∀x ∀y P(x, y)
(h)
∃x ∃y Q(x, y)
(i)
∀x ∀y ¬S(x, y)
