Exam 1: A: the Foundations: Logic and Proofs

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

express the negation of the statement in terms of quantifiers without using the negation symbol. - $\forall x ( ( x > - 1 ) \vee ( x < 1 ) )$

suppose the variables x and y represent real numbers, and $L ( x , y ) : x < y \quad Q ( x , y ) : x = y \quad E ( x ) : x \text { is even } \quad I ( x ) : x \text { is an integer }$ Write the statement using these predicates and any needed quantifiers. -Every integer is even.

Determine whether the compound propositions are satisfiable. - $( p \rightarrow q ) \wedge ( q \rightarrow \neg p ) \wedge ( p \vee q )$

(a) Find a proposition with the truth table at the right. (b) Find a proposition using only $p , q , \neg$ , and the connective $V$ that has this truth table.

suppose the variable x represents students, y represents courses, and T(x, y) means "x is taking y." Match the English statement with all its equivalent symbolic statements in this list: 1. \existsx\forallyT(x,y) 2. \existsy\forallxT(x,y) 3. \forallx\existsyT(x,y) 4. \neg\existsx\existsyT(x,y) 5. \existsx\forally\negT(x,y) 6. \forally\existsxT(x,y) 7. \existsy\forallx\negT(x,y) 8. \neg\forallx\existsyT(x,y) 9. \neg\existsy\forallxT(x,y) 10. \neg\forallx\existsy\negT(x,y) 11. \neg\forallx\neg\forally\negT(x,y) 12. \forallx\existsy\negT(x,y) -There is a course that no students are taking.

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

determine whether the proposition is TRUE or FALSE. -1 + 1 = 3 if and only if 2 + 2 = 3.

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 in good English. Do not use variables in your answer. - $\neg \exists x ( A ( x ) \wedge T ( x ) )$

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

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. - $\exists y \neg S ( \text { Margaret, } y )$

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: $S ( x , y ) : x \text { saw } y \quad L ( x , y ) : x \text { liked } y A ( y ) : y \text { won an award } \quad C ( y ) : y \text { is a comedy. }$ -Ben has never seen a movie that won an award.

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

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement. -P(1, −1)

Find a proposition with three variables $\gamma$ , and $p , q$ that is never true.

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 comedy won an award.

A student is asked to give the negation of "all bananas are ripe." (a) The student responds "all bananas are not ripe." Explain why the English in the student's response is ambiguous. (b) Another student says that the negation of the statement is "no bananas are ripe." Explain why this is not correct. (c) Another student says that the negation of the statement is "some bananas are ripe." Explain why this is not correct. (d) Give the correct negation.

Express r ⊕ d in English, where r is "it is rainy" and d is "it is dry."

suppose the variables x and y represent real numbers, and $L ( x , y ) : x < y \quad G ( x ) : x > 0 \quad P ( x ) : x \text { is a prime number. }$ Write the statement in good English without using any variables in your answer. - $\forall x \exists y L ( x , y )$

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