Problem 1: Determine whether or not the following two statements, (a) and (b), are logically equivalent. (You may use truth tables or logical properties but must use some form of formal argument for logical equivalence. Make sure to clarify the meaning of any symbols you introduce.)

(a): Parvati is studying calculus and linear algebra and Quinn is studying calculus, but Quinn is not studying both calculus and linear algebra.

(b): It is not the case that both Parvati and Quinn are both studying calculus and linear algebra, but it is the case that Quinn is studying calculus and Parvati is studying both calculus and linear algebra.

Hint: It is possible to write these using only three statement variables.

Problem 2: Use the logical equivalences established in Theorem 2.1.1 as well as the properties of conditional statements to prove that (pq)(qp)p . (Cite each equivalence used, and only use one at a time.)

Problem 3: Show that the three statements below are all logically equivalent (using equivalences in the same way as Problem 2).

(a) pqr

(b) pqr

(c) prq

Problem 4: Suppose n is a fixed integer. Use the result of Problem 3 (not other equivalences) to write another statement logically equivalent to the statement "If n is prime, then n is even or n3". (Write these in the same English style as the given statement.)

Problem 5: Suppose we know that pqr is false. From this, can we determine the truth value of the statement qppr? Explain why or why not, and determine the truth value of that statement if possible.

Note: We define the biconditional here by pq(pq)(qp).

Problem 6: For the statement "Justifying all statements and not using undefined variables is a necessary condition to have a valid proof":

(a) Rewrite this statement in a standard if-then form (in an English structure like the original).

(b) Write the negation of this statement.

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