Chapter 3: Logic

Conditional statement: P: F; Q: F =

True

Some are not

all are

Quantifier

all, no, some

None are

some are

All are

some are not

A sentence that can be judged either true or false is called​ a(n)

statement

Statement: If I don't spend all my allowance or I save some extra money, then I can afford to go on the trip. a. Which symbolic statement best represents the written​ statement? p: I spend all my allowance. q: I save some extra money.

( ~p v q) → r

Statement: If you eat at a restaurant then you will not order a dessert​, if and only if it is not a weekend. ​a) Which symbolic statement best represents the written​ statement? p: You eat at the restaurant. q: You will order a dessert. ​r: It is a weekend.

(p → ~q)↔p~r

Let​ p, q, and r represent the following simple statements. ​p: It is time to sleep. ​q: I work hard. ​r: The job pays well. It is time to sleep and I work hard​, or the job pays well. The symbolic form

(p∧q)∨r.

Let​ p, q, and r represent the following statements. ​p: The taxes are high. ​q: The job pays well. ​r: I work hard. If I work hard and the job pays well​, then the taxes are high. The symbolic form is...

(r∧q)→p

Let p and q represent the following statements. ​p: I study. ​q: I pass the class. Write the compound statement​ ~p ↔ ​~q in words.

If I do not study if and only if I do not pass the class

Truth tables

A device used to determine when statements are true or false

Euler Diagram

A diagram consisting of two circles to represent a conditional statement with the inside circle representing the hypothesis and the outside circle re[resenting the conclusion

syllogistic argument

A form of deductive reasoning in which conclusion supported by major and minor premise.

De Morgan's Law

A set of rules for converting an expression containing NOTs into an expression that does not contain any NOTs.

Determine the truth value of the statement (p ∧ ~q) ∧ r using the following conditions. ​a) p is false​, q is true​, and r is true. ​b) p is false​, q is false​, and r is false. ​b) If p is false​, q is false​, and r is false​, what is the value of (p ∧ ~q) ∧ r​?

False

Write the negation of the statement: Some crabs do not have claws.

All crabs have claws

Determine the truth value of the statement (p ∨ ~q) ∧ r using the following conditions. ​a) p is false​, q is true​, and r is false. ​b) p is true​, q is true​, and r is true. ​a) If p is false​, q is true​, and r is false​, what is the truth value of (p ∨ ~q) ∧ r​?

False

Determine the truth value of the statement (~r ∧ ~p) ∧ q using the following conditions. ​a) p is false​, q is false​, and r is false. ​b) p is false​, q is true​, and r is true. a) If p is false​, q is false​, and r is false​, what is the value of (~r ∧ ~p) ∧ q​?

False

Use De​ Morgan's laws to write an equivalent statement for the given sentence. It is false that Australia is an island or Mexico is an island.

Australia is not an island and Mexico is not an island

Use the fact that ​~(p→​q) is equivalent to p ∧ ​~q to write the statement in an equivalent form. ​Statement: It is false that if Bob went shopping​, then he ate.

Bob went shopping and he did not eat

Use De​ Morgan's laws to write an equivalent statement for the given sentence. It is false that Charles Schultz wrote a sonata or Snoopy danced a jig.

Charles Shultz did not write a sonata and Snoopy did not danced a jig

Use the fact that ​~(p→​q) is equivalent to p ∧ ​~q to write the statement in an equivalent form. ​Statement: It is false that if Earl went swimming​, then he got exercise.

Earl went swimming and he did not get exercise

The validity of a syllogistic argument can be determined using​ a(n) _______ diagram.

Euler

Determine the truth value of the statement (~r ∧ ~p) ∧ q using the following conditions. ​a) p is false​, q is false​, and r is false. ​b) p is false​, q is true​, and r is true. b) If p is false​, q is true​, and r is true​, what is the value of (~r ∧ ~p) ∧ q​?

False

Use De​ Morgan's laws to determine whether the two statements are equivalent. ~(a ∨ ~b), ~a ∨ b

The two statements are not equivalent.

Determine the truth value of the statement (p ∧ ~q) ∧ r using the following conditions. ​a) p is false​, q is true​, and r is true. ​b) p is false​, q is false​, and r is false. a) If p is false​, q is true​, and r is true​, what is the value of (p ∧ ~q) ∧ r​?

False

If p is true​, q is false​, and r is true​, find the truth value of the statement. (~p ↔ r) ∧ (~q ↔ r) Select the truth value of (~p ↔ r) ∧ (~q ↔ r) when p is true​, q is false​, and r is true.

False

If p is true​, q is false​, and r is true​, find the truth value of the statement. (~p ↔ r) ∧ (~q ↔ r) Select the truth value of (~p ↔ r) ∧ (~q ↔ r) when p is true​, q is false​, and r is true. Choose the correct answer below. True

False

If p is true​, q is false​, and r is true​, find the truth value of the statement. (~p ∧ ~q) ∨ ~r Select the truth value of (~p ∧ ~q) ∨ ~r when p is true​, q is false​, and r is true.

False

The disjunction p ∨ q is false only when both p and q are​

False

If p is false​, q is true​, and r is true​, find the truth value of the statement. (p ∧ q) ↔ (q ∨ ~r)

False because (p ∧ q) is false and (q ∨ ~r) is true.

Use the fact that ​~(p→​q) is equivalent to p ∧ ​~q to write the statement in an equivalent form. ​Statement: It is false that if Felix had his hair cut​, then he went to the concert.

Felix had his hair cut and he did not go to the concert.

Use the fact that ​~(p→​q) is equivalent to p ∧ ​~q to write the statement in an equivalent form. ​Statement: It is false that if Gerald ate lunch​, then he got enough nutrition.

Gerald ate lunch and did not get enough nutrition

Use De​ Morgan's laws to write an equivalent statement for the given sentence. It is false that he does want to smile or he is mad.

He does not want to smile and he is not mad.

Use the fact that p→q is equivalent to​ ~p ∨ q to write an equivalent form of the following statement. ​Statement: Angel set the alarm clock or he did not wake on time.

If Angel did not set the alarm clock​, then he did not wake on time.

Use the fact that p → q is equivalent to​ ~p ∨ q to write an equivalent form of the given statement. Chase is not hiding or the pitcher is broken

If Chase hiding, then pitcher is broken

Use the fact that p → q is equivalent to ~p ∨ q to write an equivalent form of the given statement. Chase is not hiding or the pitcher is broken.

If Chase is hiding, then the pitcher us broken

Let​ p, q, and r represent the following simple statements. ​p: The job pays well. ​q: I get an A. ​r: The stove is hot. Write the symbolic statement in words. (q → r) ∧ p

If I get an A then the stove is hot, and the job pays well

Use the fact that p→q is equivalent to​ ~p ∨ q to write an equivalent form of the following statement. ​Statement: Idaho is a state or the toy was not made in the USA.

If Idaho is not a state​, then the toy was not made in the USA.

Fallacy of the Converse

If P then Q Q Therefore P A conditional and its converse are not​ equivalent, so it is an example of the fallacy of the converse.

Law of Detachment

If a conditional is true and its hypothesis is true, then its conclusion is true.

The contrapositive for the following statement. If there must be an early worm, then the birds flock together.

If the birds do not flock together, then there must not be an early worm

The converse for the following statement. If there must be an early worm, then the birds flock together.

If the birds flock together, then there must be an early worm

Let​ p, q, and r represent the following simple statements. ​p: The stove is hot. ​q: I study. ​r: The chair is broken. Write the symbolic statement in words. ~r → (q ∧ p)

If the chair is not broken then I study, and the stove is hot.

The inverse for the following statement. If there must be an early worm, then the birds flock together.

If there must not be an early worm, then the birds do not flock together

Let p and q represent the following statements. ​p: This is an octopus. ​q: The aquarium is full of fish. Write the compound statement​ ~p → q in words.

If this is not an octopus, then the aquarium is full of fish

Write the​ converse statement. If you mow the lawn, then you can go play baseball.

If you can go play baseball, then you can mow the lawn

Write the​ contrapositive of the statement. If you mow the lawn, then you can go play baseball.

If you cannot go play baseball, then you did not mow the lawn.

inverse of the statement. If you mow the lawn, then you can go play baseball.

If you did not mow the lawn, then you cannot go play baseball.

Let p represent the following statement. ​p: Interest is a payment for the use of borrowed money. Express the symbolic statement​ ~p in words

Interest is not a payment for the use of borrowed money.

Determine whether the argument is valid or invalid. You may compare the argument to a standard form or use a truth table. a ∨ b a ... b Is the argument valid or​ invalid?

Invalid

Determine whether the argument is valid or invalid. You may compare the argument to a standard form or use a truth table. m ∨ n m ... n Is the argument valid or​ invalid?

Invalid

Determine whether the argument is valid or invalid. You may compare the argument to a standard form or use a truth table. p → q p ... ~q Is the argument valid or​ invalid?

Invalid

Use an Euler diagram to determine whether the syllogism is valid or invalid. All swimmers float. Squeaky is not a swimmer. therefore Squeaky does not float

Invalid

Fallacy of the Inverse

It is an invalid​ argument, because a conditional and its inverse are not equivalent. If P then Q P Therefore Q

Use the fact that ​~(p→​q) is equivalent to p ∧ ​~q to write the statement in an equivalent form. ​Statement: Dan visited the museum and he did not see the painting.

It is false that if Dan visited the museum​, then he saw the painting.

Use the fact that ​~(p→​q) is equivalent to p ∧ ​~q to write the statement in an equivalent form. Statement: Earl went swimming and he did not get exercise.

It is false that if Earl went swimming​, then he got exercise.

Use the fact that ​~(p→​q) is equivalent to p ∧ ​~q to write the statement in an equivalent form. ​Statement: Felix had his hair cut and he did not go to the concert.

It is false that if Felix had his hair cut​, then he went to the concert.

Use the fact that ​~(p→​q) is equivalent to p ∧ ​~q to write the statement in an equivalent form. ​Statement: Harry runs quick and he did not catch Sally.

It is false that if Harry runs quick ,then he caught Sally.

Use the fact that ​~(p→​q) is equivalent to p ∧ ​~q to write the statement in an equivalent form. ​Statement: Jerry caught the train and he did not get home.

It is false that if Jerry caught the train​, then he got home.

Use De​ Morgan's laws to write an equivalent statement for the given sentence. The bus has an engine​, but the people do not have money

It is false that the bus has an engine​ or people have money

Use De​ Morgan's laws to write an equivalent statement for the given sentence. The hotel has a bedroom​, but the conference center does not have a projector.

It is false that the hotel has a bedroom​ or the conference center has a projector.

Use De​ Morgan's laws to write an equivalent statement for the given sentence. The hotel does not have a weight room and the conference center does not have an auditorium

It is false that the hotel has a weight room or the conference center has an auditorium

Use De​ Morgan's laws to write an equivalent statement for the given sentence. The party has a cake​, but the people do not have a celebration.

It is false that the party does not have a cake​, or the people have a celebration

Use De​ Morgan's laws to write an equivalent statement for the given sentence. The party does not have a cake and the people do not have

It is false that the party has a cake or the people have a celebration

Use De​ Morgan's laws to write an equivalent statement for the given sentence. The rocket has wings​, but the launchpad does not have a stand

It is false that the rocket has wings​ or the launchpad has a stand

Use De​ Morgan's laws to write an equivalent statement for the given sentence. The store does not have a bathroom and the park does not have an ice skating rink

It is false that the store has a bathroom or the park has an ice skating rink

Use De​ Morgan's laws to write an equivalent statement for the given sentence. It is false that it is Halloween or goblins come out to play.

It is not Halloween and the goblins do not come out and play

Let​ p, q, and r represent the following simple statements. ​p: It is Tuesday. ​q: It is snowing outside. ​r: The taxes are high. ~p ∧ (q ∨ r) The symbolic statement in words...

It is not Tuesday, and it is snowing outside or the taxes are high

Let p and q represent the following simple statements. ​p: The job pays well. ​q: The play is boring. Write the symbolic statement ~(p∧q) in words.

It is not true that the job pays well and the play is boring

Let p and q represent the following simple statements. ​p: I eat bananas. ​q: The stove is hot. Write the symbolic statement ~(q∨p) in words.

It is not true the stove is hot and I eat bananas

Use the fact that p → q is equivalent to ​ ~p ∨ q to write an equivalent form of the given statement. If Janette buys a new house​, then she sells her old house.

Janette does not buy a new house or she she sells her old house

Use the fact that p → q is equivalent to​ ~p ∨ q to write an equivalent form of the given statement. If Janette buys a new lawnmower​, then she sells her old lawnmower.

Janette does not buy a new lawnmower or she sells her old lawnmower

Use the fact that p → q is equivalent to​ ~p ∨ q to write an equivalent form of the given statement. If Janette takes a new job​, then she leaves her old job.

Janette does not take her new job, or she leaves her old job

Use the fact that p → q is equivalent to​ ~p ∨ q to write an equivalent form of the given statement. If Janette buys a new bicycle​, then she sells her old bicycle.

Jannette does not buy a new bicycle or she does not sell her her old bicycle

The Barr triplets have an annoying​ habit: Whenever a question is asked of the three of​ them, two tell the truth and the third lies. When asked which of them was born​ last, they replied as follows. Mary​ said, "Katie was born​ last." Katie​ said, "I am the​ youngest." Annie​ said, "Mary is the​ youngest." Which of the Barr triplets was born​ last? Choose the correct answer below.

Katie

The five basic Truth tables

Negation, Conjunction, disjunction, conditional, and biconditional

Write the negation of the statement: Some cars are on the road

No cars are on the road

What conclusion would make the following syllogism​ valid? No frogs are mammals. All dolphins are mammals. therefore. ________

No frogs are dolphins

conditional truth table

P Q P>Q T T T T F F F T T F F T

Disjunctive Syllogism

P or Q Not P Therefore Q

Write the negation of the statement. All turkeys fly.

Some Turkeys can fly

Write the negation of the statement: No novels have over 500 pages

Some novels have over 500 pages

Write the negation of the statement: No shopping carts have three wheels.

Some shopping carts have three wheels

Equivalent Statements

Statements that have exactly the same truth values in the answer columns of their truth tables

p → q ​~p therefore ~q Is this valid or invalid?

The argument is invalid because its corresponding statement is not a tautology.

Determine whether the argument is valid or invalid. You may compare the argument to a standard form or use a truth table. a → b ~b therefore a Is the argument valid or​ invalid?

The argument is invalid because the conclusion is not a tautology.

Use De​ Morgan's laws to write an equivalent statement for the given sentence. It is false that the band plays rock music and the band has a guitarist.

The band does not play rock music or the band does not have a guitarist

Let p and q represent the following statements. ​p: The belt is brown. ​q: The hat is tan. Write the compound statement p ∨ ~q in words.

The belt is brown or the hat is not tan

What is true about the biconditional statement p↔​q?

The biconditional statement is true only when p and q have the same truth value.

Determine the truth value for each simple statement.​ Then, using the truth​ values, determine the truth value of the compound statement. President's Day is in March and Memorial Day is in August​, if and only if Thanksgiving is in November.

The compound statement is false because ​"President's Day is in March and Memorial Day is in August​" is false and ​"Thanksgiving is in November​" is true.

What is correct about the conditional statement p→​q?

The conditional statement is true in every case except when p is true and q is false.

Use a truth table to determine whether the statement is a​ tautology, self-contradiction, or neither. ~q∧​(~p∧q​) Is the statement ~q∧​(~p∧q​) a​ tautology, self-contradiction, or​ neither?

The correct answer is Self-contradiction because the statement is always false.

Use De​ Morgan's laws to determine whether the two statements are equivalent. ~(~x ∨ ~y), x ∧ y

The two statements are equivalent

Use De​ Morgan's laws to determine whether the two statements are equivalent. ~(p ∧ q), ~p ∧ ~q

The two statements are not equivalent

Let p and q represent the following statements. ​p: The scarf is brown. ​q: The tie is red. Write the compound statement ~p ∧ q in words.

The scarf is not brown and the tie is red

Statement: A figure is a quadrilateral if and only if it has four sides.

The statement is a compound statement because it combines two or more simple statements.

Statement: Albany is a city in New York and is the capital of New York.

The statement is a compound statement because it combines two or more simple statements.

Statement: If a polygon has three sides, then it is a triangle.

The statement is a compound statement because it combines two or more simple statements.

Let​ p, q, and r represent the following simple statements. ​p: This is an octopus. ​q: It is snowing outside. ​r: The chair is broken. (p ∨ q) ∧ ~r The symbolic statement in words...

This is an octopus or it is snowing outside, and the chair is not broken.

Determine the truth value of the statement (p ∨ ~q) ∧ r using the following conditions. ​a) p is false​, q is true​, and r is false. ​b) p is true​, q is true​, and r is true. If p is true​, q is true​, and r is true​, what is the truth value of (p ∨ ~q) ∧ r​?

True

If a is true​, b is false​, and c is false​, find the truth value of the statement. a → (b → c) Select the truth value of a → (b → c) when a is true​, b is false​, and c is false.

True

If p is false​, q is true​, and r is true​, find the truth value of the statement. (~p ↔ r) ∨ (~q ↔ r) Select the truth value of (~p ↔ r) ∨ (~q ↔ r) when p is false​, q is true​, and r is true. Choose the correct answer below. True This is the correct answer.

True

If p is true​, q is false​, and r is false​, find the truth value of the statement. (~p ∧ ~q) ∨ ~r Select the truth value of (~p ∧ ~q) ∨ ~r when p is true​, q is false​, and r is false. Choose the correct answer below.

True

If p is true​, q is true​, and r is false​, find the truth value of the statement. (~p ↔ r) ∨ (~q ↔ r) Select the truth value of (~p ↔ r) ∨ (~q ↔ r) when p is true​, q is true​, and r is false.

True

If p is true​, q is true​, and r is false​, find the truth value of the statement. (~p ∨ ~q) ∨ ~r Select the truth value of (~p ∨ ~q) ∨ ~r when p is true​, q is true​, and r is false.

True

The conjunction p∧q is true only when both p and q are

True

Determine whether the argument is valid or invalid. You may compare the argument to a standard form or use a truth table. m ∨ n ​~m ... n Is the argument valid or​ invalid?

Valid

Law of Syllogism

allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement

The conjunction is symbolized by ∧ and is read​ "_______."

and

A statement that consists of two or more simple statements is called a

compound statement

To determine the validity of an argument with two​ premises, construct a truth table for a conditional statement of the form [(premise 1) ∧ (premise 2)] → ______________

conclusion

Fill in the blank to complete the sentence below. q → r r therefore q An argument of the given form is called the fallacy of the

converse

Of the​ converse, inverse, and​ contrapositive, only the contrapositive of the conditional statement is _________to the conditional statement.

equivalent

An argument that is invalid is also known as a​

fallacy

The conditional statement p → q is only (false/true) when p is true and q is false.

false

The biconditional is symbolized by ↔ and is read​ "_______."

if and only if

The conditional is symbolized by → and is read​ "_______."

if-then

Use an Euler diagram to determine whether the syllogism is valid or invalid. No Xs are Vs. No Vs are Ts. therefore No Xs are Ts.

inValid

An argument is ______ when the conclusion does not necessarily follow from the set of premises.

invalid

If the conditional statement of the form [(premise 1) ∧ (premise 2)] → conclusion is not a ​tautology, then the argument is​ a(n) ________________.

invalid

Use an Euler diagram to determine whether the syllogism is valid or invalid. All Xs are Ys. No Ys are Zs. therefore No Xs are Zs.

invalid

Use an Euler diagram to determine whether the syllogism is valid or invalid. All babies cry. Sophie is not a baby. therefore symbol Sophie does not cry. Is this syllogism valid or​ invalid?

invalid

Use an Euler diagram to determine whether the syllogism is valid or invalid. All forks are utensils. All spoons are utensils. therefore symbol All spoons are forks. Is the syllogism valid or​ invalid?

invalid

Use an Euler diagram to determine whether the syllogism is valid or invalid. No lawn weeds are flowers. Sedge is not a flower. therefore symbol Sedge is a lawn weed.

invalid

If an Euler diagram can be drawn in a way in which the conclusion does not necessarily follow from the​ premises, the syllogistic argument is​ a(n) ____________ argument.

invalid argument

Fill in the blank to complete the sentence below. p → q ~p therefore ~q An argument of the given form is called the fallacy of the _____________

inverse

Identify the standard form of the following argument. p → q q → r ... (therefore) p → r

law of syllogism

Some are

none are

The negation is symbolized by​ ~ and is read​ "_______."

not

The disjunction is symbolized by ∨ and is read​ "_______."

or

disjunction truth table

p q p v q T T T T F T F T T F F F

biconditional truth table

p q p=q T T T T F F F T F F F T

Conjunction Truth Table

p q p^q T T T T F F F T F F F F

Negation Truth Table

p ~p T F F T

Let​ p, q, and r represent the following simple statements. ​p: This is an octopus. ​q: The aquarium is full of fish. ​r: There are penguins. If this is an octopus​, then the aquarium is full of fish or there are not penguins. The symbolic form is

p → (q ∨ ~r).

For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Use a truth table to determine whether the argument is valid or invalid. ​(Ignore differences in​ past, present, and future​ tense.) If he flies to Montreal, he's in Canada. He doesn't fly to Montreal. therefore He's not in Canada. ​a) Let p be​ "He flies to Montreal​" and let q be​ "He's in Canada.​" What is the argument in symbolic​ form?

p → q ​~p therefore ~q

Let p and q represent the following simple​ statements: ​p: It is snowing outside. ​q: The stove is hot. Write the following compound statement in its symbolic form. It is snowing outside and the stove is hot The symbolic form is...

p^q

Let p and q represent the following simple​ statements: ​p: This is an octopus. ​q: It is snowing outside. Write the following compound statement in its symbolic form. This is an octopus and it is not snowing outside. The symbolic form is

p∧~q.

Let​ p, q, and r represent the following statements. ​p: This is an octopus. ​q: The aquarium is full of fish. ​r: There are penguins. This is an octopus and the aquarium is not full of fish​, and there are penguins. The symbolic form...

p∧~q∧r.

The converse of p→q is _____

q→p

A compound statement that is always false is known as a​ _______.

self-contradiction

ill in the blank to complete the sentence below. p ∨ q ~p therefore q An argument of the given form is called a disjunctive ______________.

syllogism

A compound statement that is always true is known as a

tautology

Use a truth table to determine whether the statement is a​ tautology, self-contradiction, or neither. ~p∨​(q∨p​) Is the statement ~p∨​(q∨p​) a​ tautology, self-contradiction, or​ neither?

tautology

Use a truth table to determine whether the statement is a​ tautology, self-contradiction, or neither. ~p∨​(~q∨p​) Is the statement ~p∨​(~q∨p​) a​ tautology, self-contradiction, or​ neither?

tautology

The biconditional statement p ↔ q is (true/false) only when p and q have the same truth value.

true

An argument is​ _______ if the conclusion is true whenever the premises are assumed to be true.

valid

If the conditional statement [(premise 1) ∧ (premise 2)] → conclusion is a​ tautology, then the argument is​ a(n) ______ argument.

valid

Use an Euler diagram to determine whether the syllogism is valid or invalid. All mushrooms are poisonous. A morel is a mushroom. therefore symbol A morel is poisonous.

valid

Use an Euler diagram to determine whether the syllogism is valid or invalid. No soccer players are wrestlers. All midfielders are soccer players. therefore symbol No midfielders are wrestlers.

valid

If an Euler diagram can be drawn only in a way in which the conclusion necessarily follows from the​ premises, the syllogistic argument is​ a(n) ______________

valid argument

Let p and q represent the following statements. ​p: Today is Monday. ​q: The moon is made of green cheese. Express the following statement symbolically. "Today is not Monday" Symbolically the Statement is..

~ p

Statement: It is false that if you drive to school then you will not be late. a. Which symbolic statement best represents the written​ statement? p: You drive to school. q: You will be late.

~(p---> ~q)

The inverse of p → q is​ ________.

~p →~q

De​ Morgan's laws state that ~(p ∨ q) is equivalent to ______

~p ∧ ~q

De​ Morgan's laws state that ~(p ∧ q) is equivalent to ______

~p ∨ ~q

I did not buy the watch in the city and I spent $100. p: I bought the watch in the city. q: I paid $100. I did not buy the watch in the city and I spent $100. Which symbolic statement best represents the written​ statement?

~p^q

Let p and q represent the following statements. ​p: I study. ​q: I pass the class. Write the following statement in its symbolic form. I do not study​, but I pass the class. The statement in symbolic form is

~p^q

Let p and q represent the following statements. ​p: The chili is spicy. ​q: The sour cream is cold. Write the following compound statement in its symbolic form. The sour cream is not cold if and only if the chili is spicy. The statement is written...

~p~q↔p.

Let p and q represent the following simple statements. ​p: I work hard. ​q: I get a promotion. Write the following compound statement in symbolic form. I do not get a promotion if and only if I do not work hard. The compound statement in symbolic form is

~p~q↔~p.

Use letters to represent each​ non-negated simple statement and rewrite the given compound statement in symbolic form. If you do not understand your homework​, you will ask the teacher for help. Let p represent the simple sentence "You understand your homework​," and q represent ​"You will ask the teacher for help​." The compound statement written in symbolic form is

~p→q

Let p and q represent the following statements. ​p: The fourth of July is Independence Day. ​q: Salvadore Dali painted the Sistine Chapel. Express the following statement symbolically. "Salvadore Dali did no paint the Sistine Chapel."

~q

Write the statement in symbolic form. Let p and q represent the following statements. ​p: The referee is on the field. ​q: The team is in uniform. The team is not in uniform or the referee is not on the field The statement in symbolic form is...

~qv~p

Write the statement in symbolic form. Let p and q represent the following statements. ​p: The skiers are happy. ​q: The mountain is covered in snow. The mountain is not covered in snow or the skiers are not happy. The statement in symbolic form is...

~qv~p

Let p and q represent the following simple statements. ​p: This is a cat. ​q: This is a mammal. Write the following compound statement in symbolic form. If this is not a cat​, then this is not a mammal. The compound statement in symbolic form is

~q~p→~q.

Write the statement in symbolic form. Let p and q represent the following statements. ​p: The chili is spicy. ​q: The sour cream is cold. It is false that the chili is spicy or the sour cream is cold.

~​(p ∨ q)

Use the fact that ​~(p→​q) is equivalent to p ∧ ​~q to write the statement in an equivalent form. ​Statement: Carol went to the library and she read a book.

​It's false that if Carol went to the library then she did not read a book.

Determine whether the argument is valid or invalid. You may compare the argument to a standard form or use a truth table. a ∨ b a ... b ​First, write the argument horizontally. How is the argument​ translated?

​[(a ∨ b​) ∧ a​]→ b

Given the conditional statement p → q, the contrapositive of the conditional statement in symbolic form is​ __________.

​~q → ~p