Exam 1: 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.

Determine whether the following argument is valid: She is a Math Major or a Computer Science Major. If she does not know discrete math, she is not a Math Major. If she knows discrete math, she is smart. She is not a Computer Science Major. Therefore, she is smart.

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. $\forall x \exists y \neg T ( x , y )$ -Some students are taking no courses.

P(m, n) means $“ m \leq n "$ where the universe of discourse for m and n is the set of nonnegative integers. What is the truth value of the statement? - $\forall m \exists n P ( m , n )$

Prove or disprove: For all real numbers x and $y , \lfloor x + \lfloor x \rfloor \rfloor = \lfloor 2 x \rfloor$

suppose the variable x represents students and y represents courses, and: U(y): y is an upper-level course M(y): y is a math course F(x): x is a freshman B(x): x is a full-time student T(x, y): student x is taking course y. Write the statement using these predicates and any needed quantifiers. -Eric is taking MTH 281.

(a) Find a proposition with the truth table at the right. (b) Find a proposition using only $p , q , \neg,$ and the connective $\vee$ that has this truth table. p q ?

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

P(x, y) means $“ x + 2 y = x y "$ where x and y are integers. Determine the truth value of the statement. -P(0, 0).

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.

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

suppose the variables x and y represent real numbers, and L(x,y):x0 P(x):x is a prime number. Write the statement in good English without using any variables in your answer. - $\forall x \exists y [ G ( x ) \rightarrow ( P ( y ) \wedge L ( x , y ) ) ]$

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What is the rule of inference used in the following: If I work all night on this homework, then I can answer all the exercises. If I answer all the exercises, I will understand the material. Therefore, if I work all night on this homework, then I will understand the material.

Determine whether the following argument is valid: p\rightarrowr q\rightarrowr \neg(p\veeq) \therefore\negr

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.

Determine whether this proposition is a tautology: $( ( p \rightarrow \neg q ) \wedge q ) \rightarrow \neg p$

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

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.

A student is asked to give the negation of "all bananas are ripe". (a) The student responds "all bananas are not ripe". Explain why the English in the student's response is ambiguous. (b) Another student says that the negation of the statement is "no bananas are ripe". Explain why this is not correct. (c) Another student says that the negation of the statement is "some bananas are ripe". Explain why this is not correct. (d) Give the correct negation.