Deductive Reasoning (Proofs)


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Deductive Reasoning (Proofs)

 

     

     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



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