Exam 1: The Foundations: Logic and Proofs

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

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

Show that the premises "Every student in this class passed the first exam" and "Alvina is a student in this class" imply the conclusion "Alvina passed the first exam".

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

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. $\forall x \exists y \neg T ( x , y )$ -There is a course that no students are taking.

Show that the premise "My daughter visited Europe last week" implies the conclusion "Someone visited Europe last week".

P(x, y) means "x and y are real numbers such that $x + 2 y = 5 "$ Determine whether the statement is true. - $\forall x \exists y P ( x , y )$

write the negation of the statement in good English. Don't write "It is not true that . . . ." -Some bananas are yellow.

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. - $A ( \text { Bill). }$

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. -Every subset of a finite set is finite.

Determine whether the following two propositions are logically equivalent: $p \rightarrow ( \neg q \wedge r ) , \neg p \vee \neg ( r \rightarrow q ) )$

suppose the variable x represents students and y represents courses, and: $F ( x ) : x \text { is a freshman } \quad A ( x ) : x \text { is a part-time student }\quad T ( x , y ) : x \text { is taking } y \text {. }$ Write the statement in good English without using variables in your answers. - $F ( \text { Mikko } )$

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

suppose the variables x and y represent real numbers, and L(x,y):x0 P(x):x is a prime number. Write the statement in good English without using any variables in your answer. - $L ( 7,3 )$

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

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

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

Prove the following theorem: n is even if and only if n2 is even.

suppose the variable x represents students and y represents courses, and: $F ( x ) : x \text { is a freshman } \quad A ( x ) : x \text { is a part-time student }\quad T ( x , y ) : x \text { is taking } y \text {. }$ Write the statement in good English without using variables in your answers. - $\neg \exists y T ( \text { Joe, } y \text { ). }$