Logic is the science of reasoning. There are a variety of logical relationships used in math.

Use the diagram along with the statements that follow to answer the questions:

Let p represent the statement

*"an animal is a corgi."*Let q represent the statement

*"an animal is a dog."*Let r represent the statement

*"an animal is a cat."*Let s represent the statement

*"a corgi is a dog."*

If an animal is a corgi, must it also be a pet?

Determine whether the statement is true or false: \enspace\enspaceIf p, then q.

Fill in the blanks with p, q, or r to make a true statement: \enspace \enspaceIf ⬚, then ⬚.

Fill in the blanks with p, q, or r to make a true statement: \enspace \enspaceIf it's not true that ⬚, then it's not true that ⬚.

What other true statements can you write from the diagram?

In the exploration, p,\,q,\,r, and s are examples of simple logic statements or **propositions**.

New statements can be formed from these logic statements. **Negation** is the opposite of a given statement. For the statement r: "an animal is a cat" the negation is "an animal is not a cat."

We can combine simple logic statements to form **compound statements**. One type of compound statement is a **conjuction** which connects two statements with the word *and*. The conjunction of p and q from the exploration is "an animal is a corgi *and* an animal is a dog."

A **disjunction** connects two statements with the word *or*. The disjunction of r and q from the exploration is "an animal is a cat *or* an animal is a dog." In a disjunction, one or both statements may be true.

A **conditional statement**, also called an if-then statement, is a type of compound statement that combines two simple statements in the form:

\text{If $p$, then $q$.}

We call p the **premise** or the **hypothesis**. The hypothesis is the assertion that begins the statement of argument, and it typically starts with the word "if".

We call q the **conclusion**. It closes out the argument and is true if the premise is true.

The **converse** of a conditional is formed by switching the place of the premise and conclusion of the conditional statement:

\text{If $q$, then $p$.}

A true conditional statement can have a false converse, and a false conditional can have a true converse. This pair has a true conditional and a false converse:

Conditional | If the shape is a square, then the shape is a rectangle. |
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Converse | If the shape is a rectangle, then the shape is a square. |

This pair has a false conditional and a true converse:

Conditional | If an animal is a dog, then it is a corgi. |
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Converse | If an animal is a corgi, then the animal is a dog. |

A **biconditional statement** is the conjunction of a conditional statement and its converse:

\text{If $p$, then $q$; and if $q$, then $p$. }

We may also see this written as:

\text{$p$ if and only if $q$}

Consider the statement "If it is sunny, then Yasmin goes to the park."

a

Identify the hypothesis of the statement.

Worked Solution

b

Identify the conclusion.

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c

Write the converse of the statement.

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d

Use the statement and its converse to write a biconditional statement.

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Idea summary

For the statements p and q\,:

Negation - not p

Conjunction - p and q

Disjunction - p or q

Conditional - if p, then q

Converse - if q, then p

Biconditional - p if and only if q

We can use a variety of symbols when working with logical statements. This can make writing logical statements more efficient.

**Negation**

This is the opposite of a statement. We write the negation of p as \sim p.

p: | The plate is dirty. |
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\sim p: | The plate is not dirty. |

Note that the negation of a negation becomes the original proposition. In symbols we could write that \sim \left(\sim p\right) is the same as (or equivalent to) p.

**Conjunction**

To join two logical statements together with the word *and*, we use a conjunction which is often symbolized as \land.

p: | The plate is dirty. |
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q: | Dinner is over. |

p \land q: | The plate is dirty and dinner is over |

**Disjunction**

To join two logical statements together with the word *or*, we use a disjunction which is often symbolized as \lor.

p \lor q: | The plate is dirty or dinner is over |
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**Conditionals**

The symbol used for the conditional connective is a right arrow \implies. Recall a conditional is in the form:

\text{If $p$ then $q$.}

p \implies q: | If the plate is dirty, then dinner is over. |
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The connective, as a right arrow, tells us that the statement can only be read one way.

**Biconditionals**

The symbol used for the biconditional connective is a left-right arrow \iff. Recall a biconditional is in the form:

p\text{ if and only if } q

p \iff q: | The plate is dirty if and only if dinner is over. |
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This implies both of the following statements are true:

If the plate is dirty, then dinner is over. |

If dinner is over, then the plate is dirty. |

"It is false that if you eat healthy foods, then you will not get sick."

a

What is the statement in symbolic form?

p: | You eat healthy foods. |
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q: | You will get sick. |

A

\sim p \implies \sim q

B

\sim (\sim p \implies q)

C

\sim (p\implies \sim q)

D

\sim p \iff q

Worked Solution

b

What type of statement is this?

A

Conditional

B

Negation

C

Disjunction

D

Conjunction

E

Biconditional

Worked Solution

Let

p: | The lemonade is sour. |
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q: | The rice is hot. |

Write each of the following statements in symbols:

a

"The rice is not hot if and only if the lemonade is sour."

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b

"If the lemonade is sour, then the rice is not hot."

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c

The lemonade is not sour, but the rice is hot."

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d

"Neither is the lemonade sour nor is the rice hot."

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e

"It is false that the lemonade is sour or the rice is hot."

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f

"It is false that if the rice is not hot, then the lemonade is sour."

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Consider the following propositions:

p: | John is having roast beef for dinner. |
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q: | John is having Yorkshire pudding for dinner. |

r: | John is having dessert for dinner. |

Determine the meaning, in words, of the following compound proposition:

\left(p\lor q\right)\land r

Worked Solution

Consider the following statements:

p: | A triangle is a right triangle. |
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q: | The sum of the squares of the legs is equal to the square of the hypotenuse. |

Write a biconditional using those statements in both words and symbols.

Worked Solution

Idea summary

We can use a variety of symbols to make writing logic arguments more efficient.

\sim | Negation | "not" |
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\land | Conjuction | "and" |

\lor | Disjunction | "or" |

\implies | Conditional | "if - then" |

\iff | Biconditional | "if and only if" |