Exam 1: The Foundations: Logic and Proofs

Prove or disprove: For all real numbers x and $y , \lfloor x y \rfloor = \lfloor x \rfloor \cdot \lfloor y \rfloor$

Using c for "it is cold" and r for "it is rainy", write "It is rainy if it is not cold" in symbols.

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

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 all courses.

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

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 comedy won an award.

Determine whether $p \rightarrow ( q \rightarrow r ) \text { is equivalent to } ( p \rightarrow q ) \rightarrow r$

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. -Some freshman is taking an advanced course.

Prove that the equation $2 x ^ { 2 } + y ^ { 2 } = 14$ has no positive integer solutions.

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 tall angry people are friendly.

P(x, y) means $“ x + 2 y = x y "$ where x and y are integers. Determine the truth value of the statement. - $\exists y P ( 3 , y )$

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

Write a proposition equivalent to $p \rightarrow q$ using only $p , q , \neg$ , and the connective $\vee$ .

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 all students are taking.

suppose the variable x represents students and y represents courses, and: U(y): y is an upper-level course M(y): y is a math course F(x): x is a freshman B(x): x is a full-time student T(x, y): student x is taking course y. Write the statement using these predicates and any needed quantifiers. -No math course is upper-level.

What is wrong with the following "proof" that -3=3 , using backward reasoning? Assume that -3=3 . Squaring both sides yields (-3)²=3² , or 9=9 . Therefore -3=3 .

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

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

Prove: if m and n are even integers, then mn is a multiple of 4.