2. Let A(x) be the statement "x has apples," C(x) be the statement "x has cherries," and B(x, y) be the statement "x is better than y," where the domain of x and y consists of all pies. Which logical expression corresponds to the following statement? "There is some pie that has apples that is better than every pie that has cherries." a. ‡xVy(A(x) ^ (C(y) → B (x, y))) b. xvy(A(x) ^ C(y) ^ B (x, y)) c. 3xVy((A(x) ^ C(y)) → B (x, y)) d. 3xVy(A(x) → (C(y) ^ B (x, y)))

Let H(x) = " plays hockey". Let B(x) = "x plays basketball". Which statement below means "If Tom plays hockey, then everyone plays basketball." O O O O O H(Tom)→ B(Tom) yH(y) → VxB(x) H(Tom) → VxB(x) H(Tom) V VxB(x) H(Tom) →x B(x)

Suppose that: • variable X can represent any creature, • the predicate tove(X) is true if and only if X is a tove, • the predicate slithy(X) is true if and only if X is slithy, and • the function action(X) is a function that returns the kind of action that X is able to perform. Write a statement in predicate logic with the following meaning: "Any tove that can gyre or gimble must also be slithy." The following symbols are provided for you to copy if you wish (though not many of them are needed in this question, and you are not limited to using only the symbols that are listed here): 3 V A V ¬ A → € e ± Answer:

Consider a function ite (for if then else) which takes three arguments: a boolean, an expression to evaluate if the boolean is true, and a second expression to evaluate if the boolean is false. a) Given the lambda expressions for boolean constants and operators, write the lambda expression for this ite function. b) Write the lambda expression for a function iteite (for if then elseif then else) which takes five arguments: a boolean, an expression to evaluate if the boolean is true, another boolean, an expression to evaluate if the second boolean is true, and a third expression to evaluate if both booleans are false.  You may assume that a correct implementation of function ite from (a) is available to you, and use it if you like.