the goals of studying mathematics is to develop the ability to
think critically. The study of critical thinking, or
reasoning, is called
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
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
- "All triangles"
is NOT a mathematical sentence - it is a phrase.
are two types of mathematical sentences:
sentence is a sentence which contains a variable.
- "x + 2 = 8" is an open sentence -- the
variable is "x."
is my favorite color." is an open
sentence-- the variable is "It."
truth value of theses sentences
depends upon the value replacing the variable.
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.
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
a vacation day and I sleep late."
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
Sentences, or statements, that
have the same truth value are said to be
("equivalent" means "equal")