Exam 1: A: the Foundations: Logic and Proofs

Using c for "it is cold," r for "it is rainy," and w for "it is windy," write "It is rainy only if it is windy and cold" in symbols.

determine whether the proposition is TRUE or FALSE. -If 2 + 1 = 3, then 2 = 3 − 1.

suppose the variable x represents students and the variable y represents courses, and $T ( x , y ) : x \text { is taking } y \quad P ( x , y ) : x \text { passed } y \text {. }$ Write the statement in good English. Do not use variables in your answers. - $\exists y \forall x T ( x , y )$

Determine whether the following argument is valid: She is a Math Major or a Computer Science Major. If she does not know discrete math, she is not a Math Major. If she knows discrete math, she is smart. She is not a Computer Science Major. Therefore, she is smart.

On the island of knights and knaves you encounter two people, A and B. Person A says "B is a knave." Person B says "At least one of us is a knight." Determine whether each person is a knight or a knave.

Determine whether this proposition is a tautology: $( ( p \rightarrow q ) \wedge \neg p ) \rightarrow \neg q$

suppose the variable x represents students and y represents courses, and: Write the statement using these predicates and any needed quantifiers. -Every student is taking at least one course.

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

suppose the variables x and y represent real numbers, and $E ( x ) : x \text { is even } \quad G ( x ) : x > 0 \quad I ( x ) : x \text { is an integer. }$ Write the statement using these predicates and any needed quantifiers. -No even integers are odd.

Give a proof by contradiction of the following: If x and y are even integers, then xy is even.

Suppose you wish to prove a theorem of the form "if p then q." (a) If you give a direct proof, what do you assume and what do you prove? (b) If you give a proof by contraposition, what do you assume and what do you prove? (c) If you give a proof by contradiction, what do you assume and what do you prove?

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement. - $\exists y \forall x P ( x , y )$

What is the negation of the propositions -Alissa owns more quilts than Federico.

Prove that the proposition "if it is not hot, then it is hot" is equivalent to "it is hot."

assume that the universe for x is all people and the universe for y is the set of all movies. Write the English statement using the following predicates and any needed quantifiers: $S ( x , y ) : x \text { saw } y \quad L ( x , y ) : x \text { liked } y A ( y ) : y \text { won an award } \quad C ( y ) : y \text { is a comedy. }$ -No one liked every movie he has seen.

suppose the variable x represents students, F(x) means "x is a freshman," and M(x) means "x is a math major." 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. - $\neg \exists x ( M ( x ) \wedge F ( x ) )$

Prove: if m and n are even integers, then mn is a multiple of 4.

suppose the variable x represents students and y represents courses, and: $M ( y ) : y$ is a math course $\quad F ( x ) : x$ is a freshman $B ( x ) : x$ is a full-time student $\quad T ( x , y ) : x$ is taking $y$ . Write the statement in good English without using variables in your answers. - $\forall x \exists y [ ( B ( x ) \wedge F ( x ) ) \rightarrow ( M ( y ) \wedge T ( x , y ) ) ]$

Find a proposition using only $p , q , \neg$ , and the connective $V$ with the truth table at the right.

Determine whether the following argument is valid: p\rightarrowr q\rightarrowr \neg(p\veeq) $\because \neg \gamma$