Exam 1: A: the Foundations: Logic and Proofs

suppose that Q(x) is "x + 1 = 2x," where x is a real number. Find the truth value of the statement. - $\exists x Q ( x )$

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

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."

Prove that $( q \wedge ( p \rightarrow \neg q ) ) \rightarrow \neg p$ is a tautology using propositional equivalence and the laws of logic.

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) -Every student is taking at least one course.

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. - $\neg \exists x \exists y ( P ( x , y ) \wedge \neg P ( y , x ) )$

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

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. -If x < y, then x is not equal to y.

suppose the variable x represents students and the variable y represents courses, and $A ( y ) : y \text { is an advanced course } F ( x ) : x \text { is a freshman } T ( x , y ) : x \text { is taking } y \quad P ( x , y ) : x \text { passed } y \text {. }$ Write the statement using the above predicates and any needed quantifiers. -No one is taking every advanced course.

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

Suppose you are allowed to give either a direct proof or a proof by contraposition of the following: if 3n + 5 is even, then n is odd. Which type of proof would be easier to give? Explain why.

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: $S ( x , y ) : x \text { saw } y \quad L ( x , y ) : x \text { liked } y A ( y ) : y \text { won an award } \quad C ( y ) : y \text { is a comedy. }$ -Lois saw Casablanca, but didn't like it.

Prove the following theorem: $n$ is even if and only if $n ^ { 2 }$ is even.

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

determine whether the proposition is TRUE or FALSE. -If 1 < 0, then 3 = 4.

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

suppose that Q(x) is "x + 1 = 2x," where x is a real number. Find the truth value of the statement. - $Q ( 2 )$

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

A set of propositions is consistent if there is an assignment of truth values to each of the variables in the propositions that makes each proposition true. Is the following set of propositions consistent? The system is in multiuser state if and only if it is operating normally. If the system is operating normally, the kernel is functioning. The kernel is not functioning or the system is in interrupt mode. If the system is not in multiuser state, then it is in interrupt mode. The system is in interrupt mode.

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.