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1. \(\forall x \exists y P(x,y)\) is false because when \(x = 4\text{,}\) there is no \(y\) which makes \(P(4,y)\) true.

2. \(\forall y \exists x P(x,y)\) is true. No matter what \(y\) is (i.e., no matter what column we are in) there is some \(x\) for which \(P(x,y)\) is true. In fact, we can always take \(x\) to be \(3\text{.}\)

3. \(\exists x \forall y P(x,y)\) is true. In particular \(x=3\) is such a number, so that no matter what \(y\) is, \(P(x,y)\) is true.

4. \(\exists y \forall x P(x,y)\) is false. In fact, no matter what \(y\) (column) we look at, there is always some \(x\) (row) which makes \(P(x,y)\) false.