A mathematical theorem is a statement that can be
proved using deductive reasoning. This theorem is
stated within a mathematical model that has
undefined terms
definitions
postulates (assumptions based on these
terms and definitions but is unproved)
Example: A point is an undefined term.
A line is a definition: shortest distance between
two points.
Parallel lines do not intersect. (a
postulate)
The theorem asserts some
relationship of the above and/or other theorems
and this assertion is called a hypothesis.
The result being proved is called the
conclusion.
Deductive reasoning is the
logic used to move from step to step, following
these rules, and using these axioms and possibly
other proven theorems (and postulates) until the
conclusion of the theorem is reached. At this
point the theorem has been
proved. Each step along the way must directly
follow from the previous step, in other words,
all steps taken must be strongly connected.
Deductive reasoning involves
the logic: if A is true and "A
implies B" is true, then B must be true as well. In
symbols we write:
if (A ^ (A → B)) → B.
Many times you might see A →
B; this is incorrect since it does not require A
to be true. However, adding A to
this as above forces the hypothesis to be true as
well (this may seem obvious.) One result of
A → B is it is a one way step, that is the
value of B does not determine A, all that we
can say, and this is very important, is if 'not B'
then 'not A'.
It cannot be emphasized
strongly enough that a proof must use strongly
connected steps along the way to prove the result.
A
demonstration,
on the other hand, is just a specific example of
the proof, and is not a proof. A proof covers
all possible examples without relying on any
specific one. Therefore to
disprove
a theorem, all you need to do is find a counter
example.
Geometry is a good place
in mathematics to learn the method of proof. Geometry tends
to be very visual and the axioms and elementary proofs tend to
be easy to grasp. The early Greeks were the first to
document their deductive method of proof. Today's proofs
follow theirs, however most have been modernized and
simplified.
Here is
an example of a proof:
First, refer to Sum
of Angles in any Triangle
A paragraph form of proof
is supplied there. A table form is supplied here. A table
form clearly lists each step of the proof and is preferred in
the early stages of learning how to prove theorems.
Step

Deductive Reasoning

Comment

Draw line segment ED
parallel to CA through point B

Parallel lines
postulates (theorems) ensure this is possible and straight edge and
compass techniques accomplish it

Here we use
theorems already proven and construction techniques well
known and established (by Euclid and others)

Angles 1 + β
+ 2 = 180°

Straight angle
definition

Here we use an
axiom (definition)

Angle 2 = α

Alternate interior
angles are equal

Proven theorem

Angle 1 = γ

Alternate interior
angles are equal

Proven theorem

γ + β
+ α + = 180°

Direct substitution
into step 2

Algebraic
substitution

This is another FREE Geometry PRINTABLE presented to you from the
Geometry section of
K12math.com