Types of Sentences
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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")