Submission instructions: Submit your answers to the following problems as a sing

Submission instructions: Submit your answers to the following problems as a single pdf file.

Computer typesetting is encouraged, but if you are handwriting your solutions on paper or on a

touchscreen device, please make sure your solutions are legible.

Problem 1.

Verify that [? → (? → ?)] → [(? → ?) → (? → ?)] is a tautology.

Problem 2.

Assuming proposition ? is true (T), determine all truth value assignments for the propositions ?,

?, and ? for which the truth value of the following compound proposition is false (F).

(? → [(¬? ∧ ?) ∨ ¬?]) ∨ [? → (¬? ∧ ?)]

Explain your reasoning.

Problem 3.

Consider each of the following arguments. If the argument is valid, identify the rule of inference

that establishes its validity. If not, explain why.

a) Andrea can program in C++, and she can program in Java. Therefore, she can program in

C++.

b) If Ron’s computer program is correct, then he’ll be able to complete his computer science

assignment in at most two hours. It takes Ron over two hours to complete his computer

science assignment. Therefore, Ron’s computer program is not correct.

c) If interest rates fall, then the stock market will rise. Interest rates are not falling. Therefore,

the stock market will not rise.

Problem 4.

Let the universe for the variables in the following statement consist of all real numbers. Negate

and simplify ∀?∀?[(|?| = |?|) → (? = ±?)].

Problem 5.

A perfect square is an integer which is also square of an integer. More formally, ? ∈ ℤ is a perfect

square if ? = ?2 for some ? ∈ ℤ.

Prove or disprove the following two claims.

a) If ? and ? are perfect squares, then the product ?? is also a perfect square.

b)If?and?are perfect squares, then the sum?+?is also a perfect square.