d. (p^(p→q))→q h. A➜(~A↔(A⇒B))

4. Write each statement in symbolic form, and use truth tables to determine if they are logically equivalent: (define any variables you use) A: If you study hard and do all homework, then you will pass the class. B: If you study hard and did not pass the class, then you did not do all your homework.

Are the following expressions logically equivalent? (present a truthtable)~p ⋁ ~q ⋁ ~ r ; ~ (p ⋀ r ⋀ q)

Check all the statements that are true: A. It is possible to define disjunction using only negations and conjunction. B. Domain-restricted universal quantification is logically equivalent to the unrestricted universal quantification of a conditional. C. (p → a) v (p → →q) is a tautology. D. The negation of a conjunction is the disjunction of the negations. E. Inverse and converse are logically equivalent. F. A conditional is false when both premise and conclusion are false. G. The contrapositive is the inverse of the converse. H. A predicate is a proposition-valued function. I. Domain-restricted existential quantification is logically equivalent to the unrestricted existential quantification of a conditional. J. (pq) ^ (pq) is a contradiction.

Determine whether the logical statement is a tautology, a contradiction or neither: (p→q) → (p v q). [Hint: Use a truth table with 4 rows; you need to check the last row; all T is a tautology: all F is a contradition; a mixture of T and F is neither.] Example statement: If adopting new software leads to savings, then we'll either adopt new software or save money. The parentheses fit in the statement like this: If (adopting new software leads to savings), then (we'll either adopt new software or save money). This interpretation gives adopting new software leads to savings = If we adopt new software, then we'll save money O a. Tautology O b. Neither Oc. Contradiction

Which of these statements is logically equivalent to "You don't Love Hamburger or you are not American if you don't love Donuts"? Select one: O a. If you love Donuts then you are American and love Hamburger O b. You don't love Donuts, or you are American and love Hamburger O c. Both Answers

For each logical argument, select all the truth assignments for the propositional variables that show the logical argument is invalid. 1) 2) r S T T r T F T F p+ -q rv ¬S FT F T :- F F F F

Use a truth table to determine whether the given compound statement is a tautology. (p ∨ q) ↔ (q ∨ p)

b) Determine if the statement below is true or false, and explain your reasoning. If two statements are logically equivalent, then so are their negations.