
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") 
