Exam 1: The Foundations: Logic and Proofs

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. $\forall x \exists y \neg T ( x , y )$ -Every student is taking at least one course.

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

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."

Find a proposition with three variables p, q, and r that is never true.

Determine whether $( p \rightarrow q ) \wedge ( \neg p \rightarrow q ) \equiv q$

Determine whether the following argument is valid. Name the rule of inference or the fallacy. $\text { If } n \text { is a real number such that } n > 2 \text {, then } n ^ { 2 } > 4 \text {. Suppose that } n \leq 2 \text {. Then } n ^ { 2 } \leq 4 \text {. }$

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

Find a proposition with three variables p, q, and r that is true when at most one of the three variables is true, and false otherwise.

Write the contrapositive, converse, and inverse of the following: You sleep late if it is Saturday.

Determine whether the premises "Some math majors left the campus for the weekend" and "All seniors left the campus for the weekend" imply the conclusion "Some seniors are math majors."

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: $S ( x , y ) : x \text { saw } y \quad L ( x , y ) : x \text { liked } y \quad A ( y ) : y \text { won an award } \quad C ( y ) : y \text { is a comedy. }$ -Ben has never seen a movie that won an award.

suppose the variables x and y represent real numbers, and L(x,y):x

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

Determine whether the following two propositions are logically equivalent: $p \vee ( q \wedge r ) , ( p \wedge q ) \vee ( p \wedge r )$

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

Construct a combinatorial circuit using inverters, OR gates, and AND gates, that produces the outputs in from input bits p, q and r. - $( ( p \vee \neg q ) \wedge r ) \wedge ( ( \neg p \wedge \neg q ) \vee r )$ Determine whether the compound propositions in 58-59 are satisfiable.

Studying is sufficient for passing.

Determine whether the following argument is valid: p\rightarrowr q\rightarrowr q\vee\negr \thereforep

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. - $\forall x \exists y L ( x , y )$ In 127-136 suppose the variable x represents students, y represents courses, and T(x, y) means "x is taking y".

The team wins if the quarterback can pass.