Exam 1: A: the Foundations: Logic and Proofs

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. -There is no largest real number.

Explain why the negation of "Al and Bill are absent" is not "Al and Bill are present."

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

What is wrong with the following "proof" that −3 = 3, using backward reasoning? Assume that −3 = 3. Squaring both sides yields $( - 3 ) ^ { 2 } = 3 ^ { 2 } \text {, or } 9 = 9 \text {. Therefore } - 3 = 3 \text {. }$

Determine whether the following argument is valid: p\rightarrowr q\rightarrowr q\vee\negr $\therefore \neg p$

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 says "I am a spy," B says "I am a spy" and C says "B is a spy."

In 110-112 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 using these predicates and any needed quantifiers. -No friendly people are angry.

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

Prove or disprove: For all real numbers $\mathcal { X }$ and $y , \lfloor xy \rfloor = \lfloor x \rfloor . \lfloor y \rfloor$

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

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

Determine whether the premises "No juniors left campus for the weekend" and "Some math majors are not juniors" imply the conclusion "Some math majors left campus for the weekend."

Prove that p → q and its converse are not logically equivalent.

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) -No student is taking any course.

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

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. - $\forall x \exists y [ G ( x ) \rightarrow ( P ( y ) \wedge L ( x , y ) ) ]$

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

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

Determine whether the following argument is valid. Name the rule of inference or the fallacy. If n is a real number such that $n > 1 , \text { then } n ^ { 2 } > 1$ Suppose that $n ^ { 2 } > 1 . \text { Then } n > 1$

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. -The empty set is a subset of every finite set.