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For each of the arguments below formalize them in

1. For each of the arguments below, formalize them in propositional logic. If the argument *is* valid identify which inference rule was used, and formulate the tautology underlying the rule. If the argument is invalid, state whether the inverse or converse error was made.

(a) All theaters sit in the back row.

George *sits* in the back row.

∴ George is a cheater.

(b) For all students x, if* x *studies discrete math, then x is good at logic.

Dawn studies discrete math.

∴ Dawn is good at logic.

(c) If the compilation of a computer program produces error messages, then the program is not correct or the compiler is faulty.

The compilation of this program does not produce error messages.

∴this program is correct and the compiler is not faulty.

(d) All students who do not do their homework and do not study the course material will not get a good course grade.

John gets a good course grade.

∴ John did his homework or studied the course material.

2. For each of the premise-conclusion pairs below, give a valid step-by-step argument (proof) along with the name of the inference rule used in each step.

(a) Premise:* {*¬*p V** q**→*r,* sV*¬*q, *¬*t,** p**→** **t,* ¬p Λ r*→** *¬s } conclusion: ¬q.

(a) Premise: {¬p*→*r Λ ¬* s, t**→**s, *u*→*¬ *p,** *¬w, u V w}, conclusion:* ^{ }*¬t V w.

(b)Premise:* {**pV**q**, q**→*r, p Λ* s**→*t, ¬r, ¬*q** →* u Λ* s},* conclusion: t.

3. Use rules of inference to show that

(a) ∀s(*r(x)**→**(S(x)** V Q(x)))*

∃*x(** -S(x))*

∴∃x *(R(x)** →*Q(x))

*(b) *∀*x(P(x)* V *Q(x))*

∀x ((¬P(x) Λ *Q(x)** →** *R(x))

∀x(¬R(x)* →**P(x))*

4. Prove that the following four statements are equivalent:

(a) n^{2}* is *odd.

(b) 1 - n is even.

(c) n^{3} is odd.

(d) n^{2} + 1 is even.

5. (a) Give a direct proof of: "If x* *is all odd integer and y is an even integer, then x + y is odd."

(b) Give a proof by contradiction of: "If it is an odd integer, then n^{2} is odd."

(c) Give an indirect proof of: "If n is an odd integer, then n + 2 is odd."

6. Is the statement "For all positive* x, *y R, if s is irrational and y is irrational then x +* y *is irrational" True of False? If true then give a proof. If False then explain why, e.g., by giving a counterexample.

7. Consider the statement concerning integers* "If m+* n *is even,** then m - n* *is even."*

(a) Give a direct proof of the statement.

(b) Give an indirect proof of the statement.

(c) Prove the statement by contradiction.

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