Exam 1: A: the Foundations: Logic and Proofs

suppose the variable x represents students and the variable y represents courses, and $A ( y ) : y \text { is an advanced course } S ( x ) : x \text { is a sophomore } \quad F ( x ) : x \text { is a freshman } T ( x , y ) : x \text { is taking } y \text {. }$ Write the statement using these predicates and any needed quantifiers. -There is a course that every freshman is taking.

Give a proof by contradiction of the following: "If n is an odd integer, then n² is odd."

suppose the variable x represents students and the variable y represents courses, and $A ( y ) : y \text { is an advanced course } F ( x ) : x \text { is a freshman } T ( x , y ) : x \text { is taking } y \quad P ( x , y ) : x \text { passed } y \text {. }$ Write the statement using the above predicates and any needed quantifiers. -Every freshman passed calculus.

In 110-112 suppose the variable x represents people, and $F ( x ) : x \text { is friendly } T ( x ) : x \text { is tall } A ( x ) : x \text { is angry. }$ Write the statement using these predicates and any needed quantifiers. -Some people are not angry.

suppose the variable x represents students and y represents courses, and: $U ( y ) : y$ is an upper-level course $\quad M ( y ) : y$ is a math course $\quad F ( x ) : x$ is a freshman $B ( x )$ : $x$ is a full-time student $\quad T ( x , y )$ : student $x$ is taking course $y$ . Write the statement using these predicates and any needed quantifiers. -Eric is taking MTH 281.

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. - $\forall x ( \neg M ( x ) \vee \neg F ( x ) )$

Determine whether the premises "Some math majors left the campus for the weekend" and "All seniors left the campus for the weekend" imply the conclusion "Some seniors are math majors."

What is the rule of inference used in the following: If I work all night on this homework, then I can answer all the exercises. If I answer all the exercises, I will understand the material. Therefore, if I work all night on this homework, then I will understand the material.

On the island of knights and knaves you encounter two people, A and B. Person A says "B is a knave." Person B says "We are both knights." Determine whether each person is a knight or a knave.

write the statement in the form "If . . . , then . . . ." -Studying is sufficient for passing.

write the negation of the statement. (Don't write "It is not true that . . . .") -I will go to the play or read a book, but not both.

determine whether the proposition is TRUE or FALSE. -If 1 + 1 = 2 or 1 + 1 = 3, then 2 + 2 = 3 and 2 + 2 = 4.

Determine whether the following argument is valid. Name the rule of inference or the fallacy. If n is a real number such that $n > 2 , \text { then } n ^ { 2 } > 4 \text {. Suppose that } n \leq 2 \text {. Then } n ^ { 2 } \leq 4 \text {. }$

write the negation of the statement in good English. Don't write "It is not true that . . . ." -All integers ending in the digit 7 are odd.

suppose the variable x represents students and y represents courses, and: $U ( y ) : y$ is an upper-level course $\quad M ( y ) : y$ is a math course $\quad F ( x ) : x$ is a freshman $B ( x )$ : $x$ is a full-time student $\quad T ( x , y )$ : student $x$ is taking course $y$ . Write the statement using these predicates and any needed quantifiers. -No math course is upper-level.

Consider the following theorem: If x is an odd integer, then x + 2 is odd. Give a proof by contradiction of this theorem.

express the negation of the statement in terms of quantifiers without using the negation symbol. - $\exists x ( 3 < x \leq 7 )$

Prove or disprove: For all real numbers $x$ and $y , \lfloor x + \lfloor x \rfloor \rfloor = \lfloor 2 x\rfloor$

Prove that the following is true for all positive integers n: n is even if and only if 3n² + 8 is even.

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