Exam 1: The Foundations: Logic and Proofs

What is the negation of the propositions in -Abby has more than 300 friends on facebook.

Prove that $\neg p \longrightarrow \neg q$ and its inverse are not logically equivalent.

assume that the universe for x is all people and the universe for y is the set of all movies. Write the statement in good English, using the predicates $S ( x , y ) : x \text { saw } y \quad L ( x , y ) : x \text { liked } y \text {. }$ Do not use variables in your answer. - $\exists y \forall x L ( x , y )$

What is the negation of the propositions in -4.5 + 2.5 = 6

suppose the variable x represents people, and $F ( x ) : x \text { is friendly } \quad T ( x ) : x \text { is tall } \quad A ( x ) : x \text { is angry. }$ Write the statement using these predicates and any needed quantifiers. -All tall people are friendly.

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. - $\exists y \forall x T ( x , y )$

suppose that $Q ( x ) \text { is “ } x + 1 = 2 x " \text {, }$ where x is a real number. Find the truth value of the statement. - $\forall x Q ( x )$

suppose the variable x represents people, and $F ( x ) : x \text { is friendly } \quad T ( x ) : x \text { is tall } \quad A ( x ) : x \text { is angry. }$ Write the statement in good English. Do not use variables in your answer. - $\neg \exists x ( A ( x ) \wedge T ( x ) )$

write the negation of the statement in good English. Don't write "It is not true that . . . ." -No tests are easy.

suppose the variable x represents students and y represents courses, and: $M ( y ) : y$ is a math course $\quad\quad F ( x ) : x$ 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. - $\exists x \forall y T ( x , y )$

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

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

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

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

What is the negation of the propositions in -A messaging package for a cell phone costs less than $20 per month.

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.

Give a direct proof of the following: "If x is an odd integer and y is an even integer, then x + y is odd".

Given any 40 people, prove that at least four of them were born in the same month of the year.

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 $A ( x ) : x$ is a part-time student $\quad T ( x , y )$ : student $x$ is taking course $y$ . Write the statement using these predicates and any needed quantifiers. -Every part-time freshman is taking some upper-level course.

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. - $\neg P \text { (Wisteria, MAT } 100 ) \text {. }$