INDUCTIVE AND DEDUCTIVE REASONING

Simple logic was a favorite topic in my Philosophy degree, so my enthusiasm for the Geometry chapters that deal with reasoning can be easily explained.  The roots that are developed through the Geometry curriculum can be used in many other contexts: debate, essay writing, solving problems, and even convincing your parents to extending curfew.  In Geometry class right about now, if logic precedes introduction of two-column proofs, we should be thinking about inductive and deductive reasoning.

INDUCTIVE REASONING goes from the specific to the general, from observed details to global conjectures.  It’s what a scientist would use to develop a hypothesis after collecting data, discovering a pattern, and assuming that the pattern will continue.  It’s also the thinking that is expected on the last couple of questions in any data set on the ACT Science Reasoning section.

A determining factor of inductive reasoning is that conjectures based on an infinite universe can never be proven.  We can only DISPROVE them.  Let’s say a statistician collects a huge amount of data about  monkeys in a maze.  The data shows that every time a monkey consumed carrots before running the maze, the elapsed time was shorter than without eating carrots.  This pattern continued with tens of thousands of monkeys.  The scientist hypothesizes that eating carrots will always improve a monkey’s speed in running a maze.  It would be impossible to test every monkey, so we could never prove the hypothesis is universally true.  But find just one monkey whose performance did NOT improve and the hypothesis is disproven.

The same principle of finding a counterexample will be helpful later when we get to indirect proof.

DEDUCTIVE REASONING (general to specific) starts with a global statement, properties, definitions, postulates, or theorems and applies them to form conclusions in specific situations.  It assumes that what is true of the large group is true of all of the members of the group.  This is the form of reasoning we’ll be using in two-column proof in Geometry.

Within deductive reasoning, we find two important principles: the laws of DETACHMENT and SYLLOGISM.

                 DETACHMENT follows my proposal that mathematicians aren’t poets engaging in metaphors.  When mathematicians mean detach, they say detach.  So the Law of Detachment says that something can be removed or detached.

If P implies Q
and P is true (given)
then Q is also true.

Q can be detached from the original statement and concluded to be true. This is a primary construction in Geometry  proofs.

1.  If a polygon has three sides, then it is a triangle.
2.  This polygon has three sides.
3.  Therefore, it is a triangle.

                  SYLLOGISM:  If you learned the transitive property in Algebra, you already know about syllogisms.

The placement of P, Q, and R are important in syllogisms.  R is the conclusion you want to come to and is considered the MAJOR term.  P is called the MINOR term, and Q is the MIDDLE term.  As “transitive” implies, we’re moving over or across the Q to get P to relate to R.



VALIDITY AND SOUNDNESS

CAUTION!  When we’re talking about logical steps to conclusions, we need to distinguish between validity and soundness.  If an argument is VALID, it follows the syllogistic steps already listed.  Validity is determined by the form of the argument and says nothing about truth, either of the premises or the conclusion.  A conclusion can be valid but unsound.

If the premises are not true, even if they follow appropriate steps, then the conclusion is not sound.  In order for an argument to be SOUND, the proper logical steps must be followed and premises must also be true. 

Here’s an example of a valid argument which is NOT sound.

    All my pets are mammals.
    Lizzie the Lizard is my pet.
    Therefore, Lizzie the Lizard is a mammal.  (Valid, but false: a lizard is not a mammal. The conclusion is not sound because the first premise is not true.)

Some students try to determine the validity of a conclusion by deciding if it and the premises are true.  This is a slippery slope in logic.  A better approach is to put premises into the “P implies Q” format and look at the steps used to arrive at the conclusion.

This example is not even valid because the steps are wrong, but the conclusion could be either true or false:

P, Q, and R statements might go like this:
    All dogs are mammals.
    All cats are mammals.
    All dogs are cats.  (Not Valid because the steps are wrong, and not Sound even though both premises are true, and obviously false.)

Looking only at the truth of the conclusion to determine its soundness can lead to mistakes.
    All mammals are born alive.
    All elephants are born alive.
    Therefore, all elephants are mammals.  (The conclusion is true, but the logic is not valid because it does not follow the Law of Syllogism, and the conclusion is not sound because the first premise is not true.  The platypus is not a mammal but offspring are born alive.)

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SUMMARY FOR YOUR LOGIC STUDY CARD:

Inductive Reasoning: specific to general

Deductive Reasoning: general to specific
     Detachment: If A, then B.  A.  Therefore, B.
     Syllogism:  If A, then B.  If B, then C.  Therefore, If A, then C.

Validity:  Are the Laws of Detachment and Syllogism followed?

Soundness:  Are the premises valid AND true?

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.

BASICS OF STATISTICS for ADVANCED ALGEBRA STUDENTS

Some of the concepts in your first Statistics test are already part of your mathematical skill set.  You already know, for example, how to calculate average or mean, median, and mode.  These are measures of central tendency.  They tell typical values like (respectively) the score that everyone would have gotten on that test if all points earned had been evenly distributed among the students, the middle of the scores where half of the students got a higher grade and half a lower one, or the most frequent score received.

You probably understand the concept of range - the spread between two values, the difference between the highest score on a test and the lowest.  You may even have drawn box and whiskers diagrams and frequency graphs as long ago as elementary school.

In higher level Algebra courses, we add a few details to make statistical measures and displays even more useful.

Some useful SYMBOLS

 (Pronounced "mew") This Greek letter stands for the POPULATION MEAN.

 (Called "ex bar")  This symbol represents the SAMPLE MEAN.


(Small sigma from the Greek alphabet)  Sigma is the symbol for for POPULATION STANDARD DEVIATION.

The small ess is used to signify the SAMPLE POPULATION STANDARD DEVIATION.

Using the MEDIAN

Median is used to construct a BOX and WHISKERS PLOT.

The middlemost element in a collection tells us where 50% of the subjects or items are higher and 50% lower.  The middle element (median) of the lower half and the middle element (median) of the upper half are the first and third quartiles (Q1 and Q3).

Quartiles divide the data into quarters, so at Q1, 1/4 of the scores are lower, while at Q3, 1/4 are higher.  The difference between Q1 and Q3 is the interquartile range, IQR.  It describes the location of the middle 50% of items in the group and is less influenced by extreme values than some other measures.  In fact, we use the IQR to determine whether values are too far from the central tendency to be worthy of consideration.

Say we want to know how many students are likely to attend the basketball game this weekend.  We’ve kept count of the number of attendees at the past 15 games and the interquartile range is 100.  But last month, there was one game that was attended by 700 fans!  That statistic is  so high, it could throw off measures of central tendency like the mean, so how do we tell for sure if we should count it as the highest value element or something else?

Identifying OUTLIERS

Using the IQR, we can calculate the limits beyond which certain values are too extreme to be of importance.  We call these outliers.

The limit of “reasonable” element values is 1.5 times the IQR either above Q3 or below Q1.

In our basketball game example, we can arbitrarily say that 100 is our mean.  the reasonable upper limit would be 1.5 X 100, or 150 above the Q3 count of 200 (another arbitrary example for demonstration sake), or 350 fans.  That 700 game was definitely an outlier.

On the box and whiskers plot, outliers are marked, maybe by an X but OUTSIDE the whiskers.

The measures forming the box plot (excluding any outliers) are sometimes called the “5 Number Summary.”

Using the MEAN

We’ve been calculating the arithmetic mean (average) since grammar school.  Almost every 5th grader will tell you that to figure out the average, “add ’um all up and divide by the number of ‘um.”  The use of the mean in statistics is yet another example of my guiding principle: you learned everything there is to know in math before you entered middle school.  From then on, we just use those things in new ways. 

So let’s use the mean to figure out how things are distributed around that central tendancy.  Let’s first calculate the VARIANCE by finding the distance of each value from the mean...

 then squaring it to remove the implication of more than or less than the average...

Now divide by the number of items and we have an average of the squares...

In these equations, I've used the symbols for SAMPLE MEAN.  To calculate the POPULATION MEAN, use mu.  Some texts distinguish the Sample Mean from the Population Mean by using n-1 as the denominator in the former.  I've never made that distinction, but check with your text to see what your class requires.

Either Variance equation used gives us a measure of how spread out the items are, but squaring the differences has provided a number that isn’t the same “weight” as the individual items.  To find the STANDARD DEVIATION, we need to undue those squares...

Use this equation for standard deviation when you are able to survey every member of the population.  If a smaller sample is used, errors could effect the standard deviation.  Experience has suggested a modest alteration in the equation in order to gain a more reliable result:

Use this equation for standard deviation when you only tested a sample of the entire population.  This minor alteration in the denominator was suggested after statisticians found that it reduces some of the inaccuracy created when a small sample is used to extrapolate to a full population.

Using Standard Deviation with a Normal Bell-shaped Curve

Given a normal curve (that bell-shaped graph that puts the mean in the middle along with median and mode and tapers off at both ends), we can predict that roughly 68% of the values will fall within one standard deviation of the mean, 95% within two standard deviations, and almost every value (99%) within 3 standard deviations.

Are you wondering where the percentages came from?  So did I.  It seems that statisticians for many, many years have collected and analyzed data and actually “discovered” that for things that approach a normal curve (like heights of adult males) the 68-95-99 standards are the areas under the curve at points very close to 1-2-3 standard deviations from the mean.  I’m just taking their word for it because I am not willing to do as much arithmetic as it would take to verify the data.  I’m accepting the percents as a postulate.


Using a Z score

When I want to compare two different surveys, like a student’s results on two separate Math tests, it would be handy to have a single measure of comparison.  Here’s where the Z-score comes in handy.  It tells how each score deviates from the mean of that particular test.

A Z-score of -1 is one standard deviation below the mean, while +2 means the grade was 2 standard deviations above.  To calculate the Z-score, find the difference between the score and the mean, and divide by the standard deviation...

These 5 equations are probably more than will be covered in the first Statistics test.  Much of what comes later in the course also relies on equations, so I suggest that you start a note card now, before too much information gets out of control.  

Coming up, we’ll look at correlation coefficient, confidence intervals, and probability indexes.  Go figure!

PARTIAL FRACTIONS (Revisited)

WHY DO WE STUDY PARTIAL FRACTIONS IN ALGEBRA AND PRECALCULUS?

This one is for Emma.  A "strange" question appeared on her Precalc review worksheet.  It didn't appear to be related to the transformation of graphs concept that was the focus of the homework.  The problem looked something like this:
         Solve for A and B:                    x             =       A    +    B   
                                       x^2 + 3x + 2      (x + 1)  (x + 2)       I wondered how many of the students didn't even know that this is a partial fractions question.  This is an honors class, but I don't recall the topic being covered in previous courses.  For the benefit of everyone in the class, here's the lesson we shared in our tutorial session.  (It's a repeat of the blog entry from June, 2013)

In the early days of Algebra study, we covered “collect like terms” so that the process would be a simple step once we got to FOIL (also called “double distributive” in some schools).  The same thinking encourages us to learn other basic algebraic steps before using them in more complicated mathematics.  In Calculus, we may need to decompose rational expressions before applying the rules of integration.  By doing the initial algebraic work of decomposition now, higher level concepts will be easier to learn and understand.

There are several situations which may be presented in Partial Fraction exercises.  We’ll start with the simple and work toward the more complex.

Simple rational expression:

               2x + 3       
          (x^2 - 7x +10)        Notice that the degree of the numerator is smaller than the degree of the denominator.  This is an important restriction that must be met.  (see below: “What if the Numerator is a higher degree than the Denominator?”)

    STEP 1 - factor the denominator

              2x + 3  
            (x-2)(x-5)

    STEP 2 - using the factors from the denominator, list separate fractions to form an equation. (We don’t know the numerators yet, so let’s just keep them blank until Step 3).

              2x + 3        ?          ?  
            (x-2)(x-5)= (x-2) + (x-5)

    STEP 3 - Substitute letters for the unknown numerators.

              2x + 3         A          B 
            (x-2)(x-5) = (x-2) + (x-5)

    STEP 4 - Expand the right side to have a common denominator.

              2x + 3        A(x-5) + B(x-2)
            (x-2)(x-5) =   (x-2)      (x-5)

    STEP 5 - Since denominators on both sides of the equation are the same, set the numerators equal and distribute A and B.
   
              2x + 3    =  Ax - A5 + Bx - 2B

    STEP 6 - Collect like variable terms.

              2x + 3   = (A + B)x - 5A - 2B

    STEP 7 - Create a system of equations by setting coefficients of the variables equal.

            2  = A + B
            3  = -5A -2B            The reason I prefer this ‘systems’ method to some others is that I can use the matrix function on my calculator to get answers with only a few key strokes and no manual arithmetic.
    STEP 8 - Solve the system.

            A = -7/3
            B = 13/3

    STEP 9 - Seems like a lot of steps so far, but this is the last one.  Substitute A and B values into the equation from Step 3.

              2x + 3         -7/3      13/3 
            (x-2)(x-5) =  (x-2) +   (x-5)


WHAT ABOUT MORE COMPLEX DENOMINATORS?             
   
    More factors?  Include more fractions in Step 3....
   
                 2x + 3              A         B         C  
            (x-2)(x-5)(x-1) = (x-2) + (x-5) + (x-1)

    .... and continue with Steps 4 - 9  (SOLUTION:  A = -7/3, B= 13/12, C= 5/4)

    Repeated binomial factors?  Consider the highest degree of the repeated binomial and EVERY SMALLER DEGREE to set up the equation in Step 3....

                2x + 3             A         B           C    
            (x-2)(x-5)^2   = (x-2) + (x-5) + (x-5)^2

    ....and continue with Steps 4 - 9.  (SOLUTION:  A= 7/9, B= -7/9, C=13/3)

A quadratic factor?  If the denominator is quadratic, make the numerator of the fraction a binomial...

          2x + 3             A        Bx + C   
    (x-2)(x^2 - 5)   = (x-2) + (x^2 - 5)

    ....and continue with Steps 4 - 9.  (SOLUTION: A=-7, B=7, C=16)

    What if the Numerator is a higher degree than the Denominator? 
AHA!!  Another example of how simple algebra concepts are eventually used again as steps within more complex algorithms!!  To satisfy the requirement that Partial Fractions can only be constructed if the Numerator is a lower degree than the Denominator, we need to break up the rational expression even more by LONG DIVISION.  The Partial Fraction is constructed from the REMAINDER, expressed as a fraction with the divisor as the Denominator.