Question 1

Write the contrapositive, converse, and inverse of the following: If you try hard, then you will win.

Accepted Answer

Contrapositive: If you will no

Question 2

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement.
-$$\exists y P ( 3 , y )$$

Accepted Answer

A) True 
 B)False

Question 3

Determine whether the following argument is valid. Rainy days make gardens grow. Gardens don't grow if it is not hot. It always rains on a day that is not hot. Therefore, if it is not hot, then it is hot.

Accepted Answer

The answer of Determine whether the following argument is valid....

Question 4

What is the negation of the propositions
-Abby has more than 300 friends on Facebook.

Accepted Answer

Abby has f

Question 5

P(m, n) means "m ≤ n," where the universe of discourse for m and n is the set of nonnegative integers. What is the truth value of the statement?
-$$\forall n P ( 0 , n )$$

Accepted Answer

A) True 
 B)False

Question 6

Consider the following theorem: If x is an odd integer, then x+2 is odd. Give a direct proof of this theorem

Accepted Answer

Let x = 2k + 1. Ther

Question 7

use the conditional-disjunction equivalence to ﬁnd an equivalent compound proposition that does not
involve conditions.
-$p \rightarrow ( p \wedge q )$

Accepted Answer

The answer of use the conditional-disjunction equivalence to ﬁnd an...

Question 8

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

Accepted Answer

Some part-

Question 9

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

Accepted Answer

Suppose n = 2k but n

Question 10

Prove or disprove: For all real numbers $X$ and $y , \quad \lfloor x - y\rfloor = \lfloor x\rfloor - \lfloor y\rfloor$

Accepted Answer

False: x =

Question 11

Consider the following theorem: If x is an odd integer, then x + 2 is odd. Give a proof by contraposition of this theorem.

Accepted Answer

Suppose x + 2 = 2k.

Question 12

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

Accepted Answer

The answer of Determine whether the following two propositions are...

Question 13

Prove that the equation $2 x ^ { 2 } + y ^ { 2 }$=14  has no positive integer solutions.

Accepted Answer

Give a proof by cases. There a

Question 14

use the conditional-disjunction equivalence to ﬁnd an equivalent compound proposition that does not
involve conditions.
-$\neg p \rightarrow q$

Accepted Answer

The answer of use the conditional-disjunction equivalence to ﬁnd an...

Question 15

What is the negation of the propositions
-A messaging package for a cell phone costs less than $20 per month.

Accepted Answer

A messaging package

Question 16

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

Accepted Answer

The answer of Determine whether this proposition is a tautology:...

Question 17

write the statement in the form "If . . . , then . . . ."
-A implies B.

Accepted Answer

The answer of write the statement in the form "If...

Question 18

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

Accepted Answer

The answer of suppose the variable x represents students, F(x)...

Question 19

Write a proposition equivalent to $p \rightarrow q$ using only $p , q , \neg$ , and the connective$V$.

Accepted Answer

The answer of Write a proposition equivalent to \(p \rightarrow...

Question 20

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

Accepted Answer

A) True 
 B)False

Exam 1: A: the Foundations: Logic and Proofs

Write the contrapositive, converse, and inverse of the following: If you try hard, then you will win.

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement. - ∃yP(3,y)\exists y P ( 3 , y )∃yP(3,y)

Determine whether the following argument is valid. Rainy days make gardens grow. Gardens don't grow if it is not hot. It always rains on a day that is not hot. Therefore, if it is not hot, then it is hot.

What is the negation of the propositions -Abby has more than 300 friends on Facebook.

P(m, n) means "m ≤ n," where the universe of discourse for m and n is the set of nonnegative integers. What is the truth value of the statement? - ∀nP(0,n)\forall n P ( 0 , n )∀nP(0,n)

Consider the following theorem: If x is an odd integer, then x+2 is odd. Give a direct proof of this theorem

use the conditional-disjunction equivalence to ﬁnd an equivalent compound proposition that does not involve conditions. - p→(p∧q)p \rightarrow ( p \wedge q )p→(p∧q)

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

Prove or disprove: For all real numbers XXX and y,⌊x−y⌋=⌊x⌋−⌊y⌋y , \quad \lfloor x - y\rfloor = \lfloor x\rfloor - \lfloor y\rfloory,⌊x−y⌋=⌊x⌋−⌊y⌋

Consider the following theorem: If x is an odd integer, then x + 2 is odd. Give a proof by contraposition of this theorem.

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

Prove that the equation 2x2+y22 x ^ { 2 } + y ^ { 2 }2x2+y2 =14 has no positive integer solutions.

use the conditional-disjunction equivalence to ﬁnd an equivalent compound proposition that does not involve conditions. - ¬p→q\neg p \rightarrow q¬p→q

What is the negation of the propositions -A messaging package for a cell phone costs less than $20 per month.

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

write the statement in the form "If . . . , then . . . ." -A implies B.

Write a proposition equivalent to p→qp \rightarrow qp→q using only p,q,¬p , q , \negp,q,¬ , and the connective VVV .

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

The Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

P(x, y) means "x + 2y = xy," where x and y are integers. Determine the truth value of the statement. - $\exists y P ( 3 , y )$

P(m, n) means "m ≤ n," where the universe of discourse for m and n is the set of nonnegative integers. What is the truth value of the statement? - $\forall n P ( 0 , n )$

use the conditional-disjunction equivalence to ﬁnd an equivalent compound proposition that does not involve conditions. - $p \rightarrow ( p \wedge q )$

Prove or disprove: For all real numbers $X$ and $y , \quad \lfloor x - y\rfloor = \lfloor x\rfloor - \lfloor y\rfloor$

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

Prove that the equation $2 x ^ { 2 } + y ^ { 2 }$ =14 has no positive integer solutions.

use the conditional-disjunction equivalence to ﬁnd an equivalent compound proposition that does not involve conditions. - $\neg p \rightarrow q$

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

Write a proposition equivalent to $p \rightarrow q$ using only $p , q , \neg$ , and the connective $V$ .

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