Evaluate the proposition ∀∃y.P(x,y) ⇐⇒ ∃y.∀x.P(x,y) for an unknown predicate P. Could the statement be true? Must the statement be true for all P? Give a specific P (with domains for x and y) where it is either true or false. Prove that it is true for all P, or show a contradiction.

If 1 is used to represent true and 0 is used to represent false, determine the symbolic proposition that is computed by the following function: def Proposition(p,q):if p == 1 and q == 1:return 1elif p == 1 and q == 0:return 0elif p == 0 and q == 1:return 0elif p == 0 and q == 0:return 1

A student club holds a meeting. The predicate M(x) denotes whether person x came to the meeting on time. The predicate O(x) refers to whether person x is an officer of the club. The predicate D(x) indicates whether person x has paid his or her club dues. The domain is the set of all members of the club. The names of the members and their truth values for each of the predicates is given in the following table. Indicate whether each expression is true or false. If a universal statement is not true, give a counterexample. If an existential statement is true, give an example. Name M(x) O(x) D(X) Hillary T F Bernie F F Donald F F Jeb F Carly F F Ex (M(x) A D(x)

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