Exam 1: A: the Foundations: Logic and Proofs

Suppose that you had to prove a theorem of the form "if p then q." Explain the difference between a direct proof and a proof by contraposition.

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

write the negation of the statement. (Don't write "It is not true that . . . .") -If it is rainy, then we go to the movies.

Show that the premise "My daughter visited Europe last week" implies the conclusion "Someone visited Europe last week."

write the statement in the form "If . . . , then . . . ." -x is even only if y is odd.

suppose the variable x represents students and the variable y represents courses, and $T ( x , y ) : x \text { is taking } y \quad P ( x , y ) : x \text { passed } y \text {. }$ Write the statement in good English. Do not use variables in your answers. -¬P(Wisteria, MAT 100)

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

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

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. -L(7, 3)

Show that the premises "Every student in this class passed the first exam" and "Alvina is a student in this class" imply the conclusion "Alvina passed the first exam."

Show that the premises "Everyone who read the textbook passed the exam," and "Ed read the textbook" imply the conclusion "Ed passed the exam."

What is the rule of inference used in the following:If it snows today, the university will be closed. The university will not be closed today. Therefore, it did not snow today.

Translate the given statement into propositional logic using the propositions provided: On certain highways in the Washington, DC metro area you are allowed to travel on high occupancy lanes during rush hour only if there are at least three passengers in the vehicle. Express your answer in terms of r:"You are traveling during rush hour." t:"You are riding in a car with at least three passengers." and h:"You can travel on a high occupancy lane."

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. -Some tall angry people are friendly.

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. -A(Bill)

Construct a combinatorial circuit using inverters, OR gates, and AND gates, that produces the outputs from input bits p, q and r. - $( \neg p \wedge \neg q ) \vee ( p \wedge \wedge \neg r )$

Prove that the following three statements about positive integers n are equivalent: (a) n is even; (b) $n ^ { 3 } + 1$ is odd: (c) $n ^ { 2 } - 1$ is odd

suppose the variables x 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.

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) -Some student is taking every course.