## Types of Sentences

One of the goals of studying mathematics is to develop the ability to think critically. The study of critical thinking, or reasoning, is called **logic**.

All reasoning is based on the ways we put sentences together. Let’s start our examination of logic by defining what types of sentences we will be using.

A **mathematical sentence** is one in which a fact or complete idea is expressed. Because a mathematical sentence states a fact, many of them can be judged to be “true” or “false”. Questions and phrases are **not** mathematical sentences since they cannot be judged to be true or false.

- “An isosceles triangle has two congruent sides.” is a true mathematical sentence.
- “10 + 4 = 15” is a false mathematical sentence.
- “Did you get that one right?” is NOT a mathematical sentence – it is a question.
- “All triangles” is NOT a mathematical sentence – it is a phrase.

There are **two types** of mathematical sentences:

An** open sentence** is a sentence which contains a variable.

- “
**x + 2 = 8**” is an open sentence — the variable is “**x**.” - “
**It is my favorite color.**” is an open sentence– the variable is “**It**.” - The
**truth value**of theses sentences depends upon the value replacing the variable.

A **closed sentence**, or **statement**, is a mathematical sentence which can be judged to be true or false. A closed sentence, or statement, has no variables.

- “
**Garfield is a cartoon character.**” is a**true**closed sentence, or statement. - “
**A pentagon has exactly 4 sides**.” is a**false**closed sentence, or statement.

A **compound sentence** is formed when two or more thoughts are connected in one sentence. Words such as **and, or, if…then **and** if and only** **if** allow for the formation of compound sentences, or statements. Notice that more than one truth value is involved in working with a compound sentence.

- “Today is a vacation day and I sleep late.”
- “You can call me at 10 o’clock or you can call me at 2 o’clock.”
- “If you are going to the beach, then you should take your sunscreen.”
- “A triangle is isosceles if and only if it has two congruent sides.”

Sentences, or statements, that have the same truth value are said to be **logically equivalent**. (“equivalent” means “equal”)