## Sunday, September 22, 2013

### TRUTH TABLES

In arithmetic and algebra, we are  working largely with numbers.  When Geometry comes into play, a lot of the work involves words to convey theorems and postulates.  “If a triangle has three congruent angles, then it is an equiangular triangle.”  Try writing THAT with X and Y!!  So now we need a way to work with these statements and to know when a statement is true.

You’ve already learned some things about an “if-then” conditional statement. Remember converse, inverse, and contrapositive?  For truth tables, we’ll call the first part of the conditional statement "P" and the second part "Q."  I can’t think of a specific reason for selecting these letters, but they are the ones other mathematicians use, so why not?

For a conditional statement like “If two angles form a straight line, then the angles are supplementary,”  P will be the “two angles form a straight line” part and Q will be “the angles are supplementary” part.  Notice that "if" and "then" are detached.  They always stay in the same order, but we'll move the P and Q parts around.

The original statement can be summarized as “If P, then Q.”   For further brevity, we can write
and read it as “P implies Q.”

If P is true, then not-P ( -P) is false.  And if Q is true, then not-Q (-Q) is false.
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Now for the TRUTH TABLES.  There are four ways to judge the conditional statement.  The first part (P) can be true and the second part (Q) either true or false.  Or the first part can be false while the second part is either true or false.  The truth table looks like this:

Notice that the statement is only false under one scenario:  when the first part is true but the second part is not.  That may seem counterintuitive, so let’s look at it closely.

Let’s think of a more personal conditional statement “If you miss only one question on the quiz, I will give you an A.”  If you live up to your side and I give you the A, I’ve followed through on my promise, so the first line is true.  But if you miss only one question and I renege, then Q is false and the original implication must be false also. We can all agree to the first two lines of the truth table.

But when we consider the last two lines, it’s not as obvious why the statement is still true even when P is not.  If you miss more than one question, it doesn’t matter what grade I give you, I haven’t broken my promise, so the implication has NOT been proven false and is therefore still true.

You already learned the CONVERSE, INVERSE, and CONTRAPOSITIVE of the conditional statement.  Here are the corresponding truth tables.

CONVERSE:  Q implies P.....

INVERSE:  not P implies not Q...

CONTRAPOSITIVE: not Q implies not P...

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Here’s the TRICK to accurately filling out the truth table on the next quiz.  Look at the first part of the combination.
•If it is true and the second part is also true, the conditional statement is TRUE.
*If the first part is true and the other part is false, the conditional statement is FALSE.
*If the first part is false, the truth statement is TRUE, no matter what.

In other words, the only time the conditional statement is FALSE is when the first part is true but the second is false.

You probably won't find "truth tables" outside of this unit in geometry, but you'll find them useful in unexpected ways throughout life.  Need to evaluate the speeches of political candidates or sales pitches and advertisements?  Well, if the first part of the statement isn't true in the first place, then anything in the second part can be false and still not be a technical lie.  Understand truth tables and you won't be fooled in choosing your vote or buying a product or service.  This is the logic part of math and it's a valuable lesson.  In fact, LOGIC may be a course you could take in college, especially if you major in premed or prelaw.