Exam 1: The Foundations: Logic and Proofs

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Write the truth table for the proposition $\neg ( r \rightarrow \neg q ) \vee ( p \wedge \neg r )$

Write a proposition equivalent to $\neg p \wedge \neg q \text { using only } p , q , \neg$ and the connective $\vee$

Show that the premises "Everyone who read the textbook passed the exam", and "Ed read the textbook" imply the conclusion "Ed passed the exam".

suppose the variable x represents students and y represents courses, and: U(y): y is an upper-level course M(y): y is a math course F(x): x is a freshman B(x): x is a full-time student T(x, y): student x is taking course y. Write the statement using these predicates and any needed quantifiers. -Every freshman is a full-time student.

It is hot whenever it is sunny.

let F(A) be the predicate "A is a finite set" and S(A, B) be the predicate "A is contained in B". Suppose the universe of discourse consists of all sets. Translate the statement into symbols. -No infinite set is contained in a finite set.

Consider the following theorem: If n is an even integer, then n + 1 is odd. Give a proof by contradiction of this theorem.

A set of propositions is consistent if there is an assignment of truth values to each of the variables in the propositions that makes each proposition true. Is the following set of propositions consistent? The system is in multiuser state if and only if it is operating normally. If the system is operating normally, the kernel is functioning. The kernel is not functioning or the system is in interrupt mode. If the system is not in multiuser state, then it is in interrupt mode. The system is in interrupt mode.

relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies who can either tell the truth or lie. You encounter three people, A, B, and C. You know one of the three people is a knight, one is a knave, and one is a spy. Each of the three people knows the type of person each of the other two is. For each of these situations, if possible, determine whether there is a unique solution, list all possible solutions or state that there are no solutions. - $A \text { says "I am a spy," } B \text { says "I am a spy" and } C \text { says " } B \text { is a spy." }$

Consider the following theorem: If n is an even integer, then n + 1 is odd. Give a proof by contraposition of this theorem.

In questions , determine whether the proposition is TRUE or FALSE. -1 + 1 = 3 if and only if 2 + 2 = 3.

Determine whether the following argument is valid. Name the rule of inference or the fallacy. $\text { If } n \text { is a real number such that } n > 1 \text {, then } n ^ { 2 } > 1 \text {. Suppose that } n ^ { 2 } > 1 \text {. Then } n > 1 \text {. }$

P(m, n) means $“ m \leq n "$ where the universe of discourse for m and n is the set of nonnegative integers. What is the truth value of the statement? - $\forall n P ( 0 , n )$

Prove that $p \rightarrow q$ and its converse are not logically equivalent.

Write a proposition equivalent to $p \rightarrow q$ using only $p , q , \neg$ , and the connective $\wedge$ .

Match the statement in symbols with one of the English statements in this list: 1. Some freshmen are math majors. 2. Every math major is a freshman. 3. No math major is a freshman. - $\neg \forall x ( F ( x ) \rightarrow \neg M ( x ) )$

Prove that the following three statements about positive integers n are equivalent: (a) n is even; $\text { (b) } n ^ { 3 } + 1 \text { is }$ odd; $\text { (c) } n ^ { 2 } - 1 \text { is odd. }$

Using c for "it is cold" and d for "it is dry", write "It is neither cold nor dry" in symbols.

Match the English statement with all its equivalent symbolic statements in this list: 1. $\exists x \forall y T ( x , y )$$\quad$$\quad$$\quad$ 2. $\exists y \forall x T ( x , y )$$\quad$$\quad$$\quad$ 3. $\forall x \exists y T ( x , y )$ 4. $\neg \exists x \exists y T ( x , y )$$\quad$$\quad$ 5. $\exists x \forall y \neg T ( x , y )$$\quad$$\quad$$\quad$ 6. $\forall y \exists x T ( x , y )$ 7. $\exists y \forall x \neg T ( x , y )$$\quad$$\quad$ 8. $\neg \forall x \exists y T ( x , y )$$\quad$$\quad$$\quad$ 9. $\neg \exists y \forall x T ( x , y )$ 10. $\neg \forall x \exists y \neg T ( x , y )$$\quad$ 11. $\neg \forall x \neg \forall y \neg T ( x , y )$$\quad$ 12. $\forall x \exists y \neg T ( x , y )$ -Some student is taking every course.