Chapter 2: LogicContent check — Contemporary Math, Chapter 2: Logic

Choose a lowercase letter. The most common choices are $p$, $q$, and $r$. You are not being asked to determine the truth value.

$p$: The movie Gandhi won the Academy Award for Best Picture in 1982.

Choose a lowercase letter. The most common choices are $p$, $q$, and $r$. You are not being asked to determine the truth value.

$q$: Soccer is the most popular sport in the world.

Choose a lowercase letter. The most common choices are $p$, $q$, and $r$. You are not being asked to determine the truth value.

$r$: All oranges are citrus fruits.

The negation of a logical statement has the opposite truth value of the original statement.

Original statement: Ted Cruz was not born in Texas.

Remove “not.”

Negation: Ted Cruz was born in Texas.

The negation of a logical statement has the opposite truth value of the original statement.

Original statement: Adele has a lovely singing voice.

Add “not.”

Negation: Adele does not have a lovely singing voice.

The negation of a logical statement has the opposite truth value of the original statement.

Original statement: Leaves convert carbon dioxide to oxygen during the process of photosynthesis.

Add “not.”

Negation: Leaves do not convert carbon dioxide to oxygen during the process of photosynthesis.

The negation of a logical statement has the opposite truth value of the original statement.

Original statement: $\sim p$

To negate a negative statement, remove the tilde, since the opposite of a negative is a positive.

Negation: $\sim (\sim p)=p$

The negation of a logical statement has the opposite truth value of the original statement.

Original statement: $q$

To negate an affirmative statement, add a tilde.

Negation: $\sim q$

The negation of a logical statement has the opposite truth value of the original statement.

Original statement: $r$

To negate an affirmative statement, add a tilde.

Negation: $\sim r$

“Wonder Woman is stronger than Captain Marvel” is the negation of “Wonder Woman is not stronger than Captain Marvel.”

Thus, symbolically you write $\sim (\sim p)=p$.

The statement $r$ represents “Woody and Buzz Lightyear are best friends.”

The negation $\sim r$ is translated “Woody and Buzz Lightyear are not best friends.”

No birds give live birth to their young.

All squares are rectangles. All rectangles are parallelograms, so a square is a parallelogram. All parallelograms have four sides, so a square has four sides. Thus, all squares have four sides.

You could draw a Venn diagram that shows this. Squares are a subset of rectangles. Rectangles are a subset of parallelograms. Parallelograms are a subset of things with four sides. Thus, squares are a subset of things with four sides.

All squares are parallelograms and have four sides.

You negate a “Some…not” statement by replacing “Some…not” with “All.”

Original statement: Some apples are not sweet.

Negation: All apples are sweet.

You negate a “No” statement by replacing “No” with “Some.”

Original statement: No triangles are squares.

Negation: Some triangles are squares.

You negate a “some” statement by replacing “Some” with “No.”

Original statement: Some vegetables are green.

Negation: No vegetables are green.

negation

$\sim (\sim p)=p$

The negation of a negation is the original statement.

$p$

premises

Inductive

Today we will go skiing, but last night it did not snow.

Both “and” and “but” use the conjunction connective.

The given statements:

$p$: Last night it snowed.

$q$: Today we will go skiing.

Break down the compound statement into its components.

$q\wedge \text{~}p$

Today we will go skiing if and only if it snowed last night.

The given statements:

$p$: Last night it snowed.

$q$: Today we will go skiing.

Break down the compound statement into its components.

$q$ $\leftrightarrow$ $p$

Last night it snowed or today we will not go skiing.

The given statements:

$p$: Last night it snowed.

$q$: Today we will go skiing.

Break down the compound statement into its components.

$p\vee \text{~}q$

If it snowed last night, then today we will go skiing.

The given statements:

$p$: Last night it snowed.

$q$: Today we will go skiing.

Break down the compound statement into its components.

$p$ $\to$ $p$

$\text{~}r\to (p\vee q)$

Given statements:

$p$: My roommates ordered pizza.

$q$: I ordered wings.

$r$: Our friends came over to watch the game.

Translate $\text{~}r$ as :Our friends did not come over to watch the game.

$\to$ is the conditional, which will be translated “if…then.”

$\vee$ is the inclusive or.

The comma before “then” shows that the statements in parentheses belong together.

If our friends did not come over to watch the game, then my roommates ordered pizza or I ordered wings.

$(p\wedge q)\to r$

$\wedge$ is the conjunction, which will be translated “and.”

$\to$ is the conditional, which will be translated “if…then.”

The comma before “then” shows that the statements in parentheses belong together.

Given statements:

$p$: My roommates ordered pizza.

$q$: I ordered wings.

$r$: Our friends came over to watch the game.

If my roommates ordered pizza and I ordered wings, then our friends came over to watch the game.

$\text{~}(p\vee r)$

$\sim$ is the negation symbol. Replace the symbol with “It is not the case that…”

$\vee$ is the inclusive or.

Given statements:

$p$: My roommates ordered pizza.

$q$: I ordered wings.

$r$: Our friends came over to watch the game.

It is not the case that my roommates ordered pizza or our friends came over to watch the game.

The order of logical operations is

1. Parentheses

2. Negation

3. Disjunction, conjunction

4. Conditional

5. Biconditional

For same order operations, work left to right.

The highest order operation in this expression is negation, so add parentheses around the negation.

$p\vee q\wedge \left(\text{~}r\right)$

Disjunction and conjunction are the same order, so work left to right. First evaluate the exclusive or.

$(p\vee q)\wedge \left(\text{~}r\right)$

Then finish by evaluating the conjunction.

$((p\vee q)\wedge \left(\text{~}r\right))$

The order of logical operations is

1. Parentheses

2. Negation

3. Disjunction, conjunction

4. Conditional

5. Biconditional

For same order operations, work left to right.

$\text{~}p\to q\vee r$

The highest order operation in this expression is negation, so add parentheses around the negation.

$\left(\text{~}p\right)\to q\vee r$

The next highest order operation is the disjunction. Add parentheses around the disjunction.

$\left(\text{~}p\right)\to (q\vee r)$

Finally, you can evaluate the conditional.

$(\left(\text{~}p\right)\to (q\vee r))$

The order of logical operations is

1. Parentheses

2. Negation

3. Disjunction, conjunction

4. Conditional

5. Biconditional

For same order operations, work left to right.

$\text{~}p\vee \text{~}q\leftrightarrow \text{~}(p\wedge q)$

The right side of the biconditional already has parentheses. The next step in the order of operation is negation, so add parentheses around the three negations.

$\left(\text{~}p\right)\vee (\text{~}q)\leftrightarrow (\text{~}(p\wedge q))$

The next step in the order of operations is disjunction/conjunction. Add parentheses around the disjunction.

$(\left(\text{~}p\right)\vee (\text{~}q))\leftrightarrow (\text{~}(p\wedge q))$

The lowest order operation is the biconditional. Finish by evaluating the biconditional.

$((\left(\text{~}p\right)\vee (\text{~}q))\leftrightarrow (\text{~}(p\wedge q)))$

The negation is $p\text{:}3\times 5\ne 14$

Since $3\times 5=15$, the negation is true.

The negation is $q\text{:}$ No house are built with bricks.

The negation is false since some houses are built of bricks.

The negation is $\sim r\text{:}$ Abuja is not the capital of Nigeria.

The negation is a false statement since Abuja is the capital of Nigeria.

Given statements:

$p$: Yellow is a primary color.

$p$ is true.

$q$: Blue is a primary color.

$q$ is true.

The $\wedge$ symbol represents the conjunction and is translated “and” or “but.”

A conjunction is only true when both statements are true.

$p\wedge q$

$\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}\wedge \mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}=\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}$

Given statements:

$p$: Yellow is a primary color.

$p$ is true.

$r$: Green is a primary color.

$r$ is false.

$\sim r$ is true.

$\sim r\wedge p$

A conjunction is only true when both statements are true.

$\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}\wedge \mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}=\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}$

Given statements:

$q$: Blue is a primary color.

$q$ is true.

$r$: Green is a primary color.

$r$ is false.

$q\wedge r$

A conjunction is only true when both statements are true.

$\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}\wedge \mathrm{F}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}=\mathrm{F}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}$

Given statements:

$p$: Yellow is a primary color.

$p$ is true.

$q$: Blue is a primary color.

$q$ is true.

$p\vee q$

A disjunction is only false when both statements are false.

$\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}\vee \mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}=\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}$

Given statements:

$p$: Yellow is a primary color.

$p$ is true.

$\sim p$ is false.

$r$: Green is a primary color.

$r$ is false.

$\sim p\vee r$

A disjunction is only false when both statements are false.

$\mathrm{F}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}\vee \mathrm{F}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}=\mathrm{F}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}$

Given statements:

$q$: Blue is a primary color.

$q$ is true.

$r$: Green is a primary color.

$r$ is false.

$q\vee r$

A disjunction is only false when both statements are false.

$\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}\vee \mathrm{F}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}=\mathrm{T}\mathrm{r}\mathrm{u}\mathrm{e}$

Given statements:

$p$: Yellow is a primary color.

$p$ is true.

$q$: Blue is a primary color.

$q$ is true.

$r$: Green is a primary color.

$r$ is false.

Given statements:

$p$: Yellow is a primary color.

$p$ is true.

$q$: Blue is a primary color.

$q$ is true.

$r$: Green is a primary color.

$r$ is false.

Given statements:

$p$: Yellow is a primary color.

$p$ is true.

$q$: Blue is a primary color.

$q$ is true.

$r$: Green is a primary color.

$r$ is false.

valid

A truth table for two statements has four rows of truth values.

The general rule is that for n statements there are $2^n$ rows in the truth table.

$2^2=4$

four

The truth table for one statement has two rows of truth values.

The general rule is that for n statements there are $2^n$ rows in the truth table.

$2^1=2$

two

Since you are told to assume $p$ is true and $q$ is false, you only need one row of possible truth values in your truth table.

$p\to q$

The conditional is only false when the hypothesis is true, and the conclusion is false. Using the provided truth values, you have.

If True, then False

The result is False, so write F in the last column.

Since you are told to assume $p$ is true and $q$ is false, you only need one row of possible truth values in your truth table.

$\text{~}q$

The negation of false is true, so write T under $\text{~}q$.

$p\to \text{~}q$

The conditional is only false when the hypothesis is true, and the conclusion is false. Using $p$ and the previous column, you have.

If True, then True

The result is True, so write T in the last column.

Since you are told to assume $p$ is true and $q$ is false, you only need one row of possible truth values in your truth table.

$\text{~}p$

The negation of true is false, so write F under $\text{~}p$.

$\text{~}p\to q$

The conditional is only false when the hypothesis is true, and the conclusion is false. Using the previous two columns, you have.

If False, then False

The result is True, so write T in the last column.

Since there are two statements, you need four rows for all possible combinations.

$\text{~}p$

The negation of a statement has the opposite truth value. Write the opposite value of $p$ in this column.

$\text{~}p\vee q$

A disjunction is only false when both statements are false. Write F if both the $\text{~}p$ column and $q$ column have an F. Otherwise, write T.

$q\to (\text{~}p\vee q)$

A conditional is only false when the premise is true, and the conclusion is false. Write F if the $q$ column is T and the previous column is F. Otherwise, write T.

The statement is valid because the last column is always true.

Since there are two statements, you need four rows for all possible combinations.

$\text{~}p$

The negation of a statement has the opposite truth value. Write the opposite value of $p$ in this column.

$q\wedge p$

A conjunction is true only when both statements are true. Otherwise, it is false. Write T if both the $q$ and $p$ columns have a T. Otherwise, write an F.

$\text{~}p\to (q\wedge p)$

A conditional is only false when the premise is true, and the conclusion is false. Write F if the $\text{~}p$ column has an T and the $q\wedge p$ column has an F. Otherwise, write T.

The statement is not valid because the last column is not all true.

Since you are told to assume $p$ is true and $q$ is false, you only need one row of possible truth values in your truth table.

$p\leftrightarrow q$

A biconditional is true when the two statements have the same truth value. Otherwise, it is false.

Since $p$ and $q$ have different truth values, the biconditional is false. Write an F in the last column.

The statement is false because there is an F in the last column.

Since you are told to assume $p$ is true and $q$ is false, you only need one row of possible truth values in your truth table.

$\text{~}q$

Since $q$ is false, $\text{~}q$ is true. Write T under $\text{~}q$.

$p\leftrightarrow \text{~}q$

A biconditional is true when the two statements have the same truth value. Otherwise, it is false.

Since $p$ and $\text{~}q$ are both true, the biconditional is true. Write a T in the last column.

The statement is true because there is a T in the last column.

Since you are told to assume $p$ is true and $q$ is false, you only need one row of possible truth values in your truth table.

$\text{~}p$

Since $p$ is true, $\text{~}p$ is false. Write F under $\text{~}p$.

$\text{~}p\leftrightarrow q$

A biconditional is true only when the statements have the same truth value. Otherwise, it is false. The premise is false, $\text{~}p$ and $q$ have the same truth value, so write T in the last column.

The statement is true because there is a T in the last column.

Since there are two statements, you need four rows for all possible combinations.

$p\wedge q$

A conjunction is true only when both statements are true. Write T when both $p$ and $q$ are true. Otherwise, write F.

$\text{~}(p\wedge q)$

The negation of a statement has the opposite truth value. Write the opposite of $p\wedge q$ in this column.

$\text{~}p$

The negation of a statement has the opposite truth value. Write the opposite of $p$ in this column.

$\text{~}q$

Write the opposite of $q$ in this column.

$\text{~}p\vee \text{~}q$

A disjunction is only false when both statements are false. Write an F where both $\text{~}p$ and $\text{~}q$ are false. Otherwise, write T.

$\text{~}(p\wedge q)\leftrightarrow (\text{~}p\vee \text{~}q)$

A biconditional is true only when both statements have the same truth value. Write T where $\text{~}(p\wedge q)$ and $\text{~}p\vee \text{~}q$ have the same truth value. Write F where they do not.

This statement is valid since the last column is always true.

Since there are two statements, you need four rows for all possible combinations.

$\text{~}p$

Write the opposite of $p$ in this column.

$q\wedge p$

A conjunction is true only when both statements are true. Write a T if both $q$ and $p$ are true. Otherwise, write an F.

$\text{~}p\leftrightarrow (q\wedge p)$

A biconditional is true only when both statements have the same truth value. Write T where $\text{~}p$ and $q\wedge p$ have the same truth value. Write F where they do not.

The statement is not valid since the last column is not always true.

Since there are two statements, you need four rows for all possible combinations.

$p\to q$

A conditional is only false when the premise is true and the hypotheses is false. Write an F if $p$ is true and $q$ is false. Otherwise, write a T.

$\text{~}p$

Write the opposite of $p$ in this column.

$\text{~}p\vee q$

A disjunction is only false when both statements are false. Write an F when both $\text{~}p$ and $q$ are false. Otherwise, write a T.

$(p\to q)\leftrightarrow (\text{~}p\vee q)$

A biconditional is true only when both statements have the same truth value. Write a T where both $p\to q$ and $\text{~}p\vee q$ are true. Otherwise, write an F.

The statement is valid since the last column is always true.

Since there are three statements, you need eight rows for all possible combinations.

$p\wedge q$

A conjunction is true only when both statements are true. Otherwise, it is false. Write a T where both $p$ and $q$ are true. Otherwise, write an F.

$(p\wedge q)\to r$

A conditional is only false when the premise is true and the hypotheses is false. Write an F if $p\wedge q$ is true and $r$ is false. Otherwise, write a T.

$\text{~}p$

Write the opposite of $p$ in this column.

$\text{~}q$

Write the opposite of $q$ in this column.

$\text{~}p\vee \text{~}q$

A disjunction is only false when both statements are false. Write an F in this column of both $\text{~}p$ and $\text{~}q$ are false. Otherwise, write a T.

$(\text{~}p\vee \text{~}q)\vee r$

A disjunction is only false when both statements are false. Write an F in this column of both $\text{~}p\vee \text{~}q$ and $r$ are false. Otherwise, write a T.

$((p\wedge q)\to r)\leftrightarrow ((\text{~}p\vee \text{~}q)\vee r)$