Exercise 15.
Suppose \(P(x,y)\) is some binary predicate defined on a very small domain of discourse: just the integers 1, 2, 3, and 4. For each of the 16 pairs of these numbers, \(P(x,y)\) is either true or false, according to the following table (\(x\) values are rows, \(y\) values are columns).
| 1 | 2 | 3 | 4 | |
| 1 | T | F | F | F |
| 2 | F | T | T | F |
| 3 | T | T | T | T |
| 4 | F | F | F | F |
For example, \(P(1,3)\) is false, as indicated by the F in the first row, third column.
Use the table to decide whether the following statements are true or false.
\(\forall x \exists y P(x,y)\text{.}\)
True
False
Not enough information
\(\forall y \exists x P(x,y)\text{.}\)
True
False
Not enough information
\(\exists x \forall y P(x,y)\text{.}\)
True
False
Not enough information
\(\exists y \forall x P(x,y)\text{.}\)
True
False
Not enough information