Exam 1: The Foundations: Logic and Proofs

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 student is taking at least one course.

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. - $\forall x \exists y [ ( B ( x ) \wedge F ( x ) ) \rightarrow ( M ( y ) \wedge T ( x , y ) ) ]$

Write a proposition equivalent to $p \vee \neg q$ that uses only $p , q , \neg$ and the connective $\wedge$

Determine whether the following argument is valid. If you are not in the tennis tournament, you will not meet Ed. If you aren't in the tennis tournament or if you aren't in the play, you won't meet Kelly. You meet Kelly or you don't meet Ed. It is false that you are in the tennis tournament and in the play. Therefore, you are in the tennis tournament.

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 variables $x \text { and } y$ 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.

Determine whether the premises "No juniors left campus for the weekend" and "Some math majors are not juniors" imply the conclusion "Some math majors left campus for the weekend."

If it is rainy, then we go to the movies.

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. -There is a part-time student who is not taking any math course.

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. - $A \text { says "I am not a knight," } B \text { says "I am not a spy," and } C \text { says "I am not a knave." }$

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. - $\neg \forall x ( F ( x ) \rightarrow A ( x ) )$

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

Give a proof by cases that $x \leq | x |$ for all real numbers x.

What is the rule of inference used in the following: If it snows today, the university will be closed. The university will not be closed today. Therefore, it did not snow today.

Explain why the negation of "Some students in my class use e-mail" is not "Some students in my class do not use e-mail".

suppose that $Q ( x ) \text { is “ } x + 1 = 2 x " \text {, }$ where x is a real number. Find the truth value of the statement. - $\exists. x Q ( x )$ In 63-70 P(x, y) means $“ x + 2 y = x y "$ where x and y are integers. Determine the truth value of the statement.

P(x, y) is a predicate and the universe for the variables x and y is {1, 2, 3}. Suppose P(1, 3), P(2, 1), P(2, 2), P(2, 3), P(3, 1), P(3, 2) are true, and P(x, y) is false otherwise. Determine whether the following statements are true. - $\neg \exists x \exists y ( P ( x , y ) \wedge \neg P ( y , x ) )$

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. $\forall x \exists y \neg T ( x , y )$ -No student is taking any course. In 137-147 suppose the variable x represents students, F(x) means "x is a freshman',' and M(x) means "x is

Consider the following theorem: "if x and y are odd integers, then x+y is even". Give a proof by contradiction of this theorem.