Medium Proof 0 - Notes

READ AND COPY DOWN THE ENLARGED TEXT (Same size as above):

Inductive reasoning is when you determine an answer by trial and error, observation, and/or experiment.  The answer MAY BE wrong but seems to be right considering everything you’ve seen.

Deductive reasoning is when you determine an answer by putting together an argument based on facts that you already know are true.  If your facts and logic are correct, the answer CANNOT be wrong.

A proof is a list of logical arguments that start from a hypothesis and end with a conclusion.  In mathematics, you show have statements related to the problem and the reasons why you know each reason is true.  These reasons are often called postulates, axioms, or theorems.

Right now, you don’t know too many postulates or theorems.  HOWEVER, you do know some algebra properties which can be used in the same way.  For instance,

2(x+3) = 10  Can be written as
2x + 6 = 10  because we know the Distributive property.

In a proof, this might look like this:

Prove:  If 2(x+3) = 10, then 2x + 6 = 10.
            Statement        Reason
1.            2(x+3)=10      Given
2.         2x + 6 = 10            Distributive Property

Seems short and a little dumb.  I call these mini-proofs.
If I wanted to continue solving the equation and continue providing reasons it might look like this:

Prove:  If 2(x-4) = 12, then 2x - 8 + 8= 12+8.
            Statement                    Reason
1.         2(x-4)=12               Given
2.         2x - 8 = 12                 Distributive Property
3.         2x - 8 + 8 = 12 + 8             Addition Property of Equality (you may have forgotten this but it’s basically the rule that let’s you add a number from both sides of an equation)
Note that the Reason in Step 3 is the reason you used to go from step 2 to Step 3.

A better proof would be: (COPY THIS DOWN - will explain more in a minute)


Prove:  If 3x + 1 = 7, then x=2.

            Statement                   Reason (from last step to current step)
1.         3x + 1 = 7                   Given
2.         3x + 1 – 1 = 7 – 1       Subtraction Property of Equality
3.         3x + 0 = 6                   Substitution
4.         3x = 6                        Additive Identity
5.         3x/3 = 6/3                  Division property of equality
6.         x = 2                          Substitution

You probably would not show all these steps if you were solving these on your own but you are actually doing them in your head.  You’ve just learned some shortcuts and don’t even think about these steps anymore.

But you know how to solve equations (or you should).  And after reviewing the list below you should know enough reasons to build your own two column-proof while solving algebraic equations.

Examples:
1x = 2
  x = 2              Multiplicative Identity

2q + 0 = 6
2q = 6              Additive Identity

5m - 3 = 2
5m - 3 + 3 = 2 + 3            Addition Property of Equality

2y + 4 = 8       
2y + 4 – 4 = 8 – 4            Subtraction Property of Equality

2y + 4 = 8
2y + 4 + -4 = 8 + -4            Addition Property of Equality

8n = 16
8n/8 = 16/8                  Division Property of Equality

5 = 6 (4x – 1)
5 = 24 x – 6                 Distribution Property

 

Continue