Exam 1: The Foundations: Logic and Proofs

Using c for "it is cold" and w for "it is windy", write "To be windy it is necessary that it be cold" in symbols.

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

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 course is being taken by all students.

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: $S ( x , y ) : x \text { saw } y \quad L ( x , y ) : x \text { liked } y \quad A ( y ) : y \text { won an award } \quad C ( y ) : y \text { is a comedy. }$ -Some people have seen every comedy.

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

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: $S ( x , y ) : x \text { saw } y \quad L ( x , y ) : x \text { liked } y \quad A ( y ) : y \text { won an award } \quad C ( y ) : y \text { is a comedy. }$ -No one liked every movie he has seen.

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. - $\neg \exists x ( M ( x ) \wedge \neg F ( x ) )$

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

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

Using c for "it is cold", r for "it is rainy", and w for "it is windy", write "It is rainy only if it is windy and cold" in symbols.

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. -Some real numbers are not positive.

On the island of knights and knaves you encounter two people, A and B. Person A says "B is a knave." Person B says "At least one of us is a knight." Determine whether each person is a knight or a knave. Exercises 51-53 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

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. -Some people are not angry.

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. -No friendly people are angry.

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. - $\exists x ( F ( x ) \wedge M ( x ) )$

Determine whether the following argument is valid. Rainy days make gardens grow. Gardens don't grow if it is not hot. It always rains on a day that is not hot. Therefore, if it is not hot, then it is hot.

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

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

In questions , determine whether the proposition is TRUE or FALSE. -If it is raining, then it is raining.