## Predicate Logic

Which of the following are *valid*?

- Quantifiers of the
*same*type (existential or universal) can be reordered without altering the meaning of the statement. - For example, let \(Q(x, y)\) be (\(y > x)\) and suppose the domain is nonnegative integers. Then the left side asserts that for every nonnegative integer, there is a larger nonnegative integer, which is true. The right side asserts that there exists a nonnegative integer greater than every other nonnegative integer, which is false. Therefore, the implication as a whole is false, and the statement is not valid.
- Suppose that the left side is true. Then there exists an \(x_0\) such that for all \(y\), \(R(x_0\), \(y)\) is true. Thus, for all \(y\), there exists an \(x\) (namely, \(x_0\)) such that \(R(x, y)\) is true. Therefore, the right side is also true, and the statement is valid.
- If it's not true that there is an element in the domain with property \(S\), then
*every*element in the domain must*not*have property \(S\). Conversely, if*every*element in the domain does*not*have property \(S\), it can't be true that some element in the domain has property \(S\).