Explain why the following are theorems of predicate logic:

1. 3.132[xS:P(x) Q(x)] [xS:P(x)] [xS:Q(x)]
2. 3.133[x {yS:P(y)} :Q(x)] [xS:P(x) Q(x)]
3. 3.134[x {yS:P(y) } :Q(x)] [xS:P(x) Q(x)]

Explain why the following propositions are theorems of predicate logic, assuming that x does not appear as a free variable in the expression (and assuming that S is nonempty):

1. 3.135 [xS:]
2. 3.136 [xS:P(x)] [xS:P(x)]
3. 3.137 [xS:P(x)] [xS:P(x)]
4. 3.138( [xS:P(x) ]) [xS:P(x)]
5. 3.139([xS:P(x)] ) [xS:P(x) ]