Exam 1: A: the Foundations: Logic and Proofs

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. - $\neg \forall x ( \neg F ( x ) \vee \neg M ( x ) )$

write the negation of the statement in good English. Don't write "It is not true that . . . ." -Some skiers do not speak Swedish.

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 not a knight," B says "I am not a spy," and C says "I am not a knave."

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. -If a person is friendly, then that person is not angry.

Determine whether $p \rightarrow ( q \rightarrow r )$ and $p \rightarrow ( q \wedge r )$ are equivalent.

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. -No infinite set is contained in a finite set.

Consider the following theorem: If n is an even integer, then n + 1 is odd. Give a proof by contraposition of this theorem.

Show that the hypotheses "I left my notes in the library or I finished the rough draft of the paper" and "I did not leave my notes in the library or I revised the bibliography" imply that "I finished the rough draft of the paper or I revised the bibliography."

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. -No freshman is a sophomore.

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

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. - $\forall x ( M ( x ) \rightarrow F ( x ) )$

Use a proof by cases to show that 27 is not the square of a positive integer.

suppose the variable x represents students and y represents courses, and: $F ( x ) : x \text { is a freshman } \quad A ( x ) : x \text { is a part-time student } T ( x , y ) : x \text { is taking } y \text {. }$ Write the statement in good English without using variables in your answers. -F(Mikko)

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. -Every subset of a finite set is finite.

suppose the variable x represents students and y represents courses, and: $U ( y ) : y$ is an upper-level course $\quad M ( y ) : y$ is a math course $\quad F ( x ) : x$ is a freshman $B ( x )$ : $x$ is a full-time student $\quad T ( x , y )$ : student $x$ is taking course $y$ . Write the statement using these predicates and any needed quantifiers. -Every freshman is a full-time student.

Write the truth table for the proposition $\neg ( r \rightarrow \neg q ) \vee ( p \wedge \neg r )$

Write a proposition equivalent to $p \vee \neg q$ that uses only $p , q , \neg$ , and the connective $\bigwedge$

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. -Not all sets are finite.

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 knight," B says "I am a knave," and C says "I am not a knave."