Deductive Reasoning

 

Learning Objective(s)

·         Identify and provide examples of deductive reasoning.

·         Use properties of numbers to construct simple logical arguments.

 

Introduction

 

Deductive reasoning is probably the most used process in all of mathematics. Anyone who has solved a logic puzzle like a Sudoku puzzle has used deductive reasoning. When we reason deductively, we use known facts to make logical conclusions that we know must be true. (We deduce one fact by putting together other facts.) This is different from inductive reasoning, which generalizes and conjectures based on observations rather than logic. Mathematicians (and all the rest of us, too) often use both inductive reasoning and deductive reasoning together.

 

Deductive versus Inductive Reasoning

 

Imagine we have recorded the results of experiments or observations, and decide that we see a pattern in the data. Or, imagine we have a problem to solve but aren't sure where to start, so we try to plug in some possible solutions to see what kinds of results we get. In these cases, we are reasoning inductively, making generalizations based on a limited number of observations. The trouble is, our generalizations might not be correct. Even with a lot of examples, it can be difficult if not impossible to be sure that there isn't at least one counterexample we just haven't found yet.

 

For that reason, mathematicians, scientists, researchers, and other people who make conjectures will often follow up inductive reasoning with some deductive reasoning. They try to justify the conjectures they made based on their observations. That is, they try to provide a logical argument, a series of verifiable statements that explains why their conjecture is always true.

 

Deductive reasoning is helpful when inductive reasoning is not appropriate, or there aren't enough examples to generalize from.

 

Consider puzzles like Sudoku. In a Sudoku puzzle, each row, column, and 3-by-3 box must be filled with the digits from 1 to 9. A digit can't be used more than once in any single row, column, or box.

 

 

Inductive reasoning is no help in figuring out this puzzle. We could look at the solutions for other puzzles, and see where the digits were placed in each one. But if we generalize this to mean that we should fill the numbers in the same way every time, we won't be able to solve many puzzles!

 

What we can do instead is figure out the specific placements for this puzzle based on the rules of the game and the numbers that are given. In the top row of 3 X 3 boxes in the puzzle above, the left and middle boxes already have a 1. The 1 in the right box can't be in the first or second rows, because they already have 1s. So the 1 for the right box has to be in the third row, and there's only one place for that. Deductive reasoning has revealed the location of the 1 in the right box!

 

 

Even without Sudoku, we've all used deductive reasoning a lot. Every time we work with an equation or mathematical expression to arrive at a conclusion or answer, we are using deductive reasoning—following the general principles of mathematics to find a specific solution that must be true.

 

Example

Problem

Terry does the following to factor the expression 3x3 + 6x2 + 3x.

 

• First he factors out 3x using the distributive property:

3x3 + 6x2 + 3x = 3x(x2 + 2x + 1)

 

• Then he recognizes that the second factor is in the form

a2 + 2ab + b2, and remembers that a2 + 2ab + b2 = (a + b)2

 

• Finally, he rewrites the second factor as the square of a binomial: 3x(x2 + 2x + 1) = 3x(x + 1)2

 

Is this inductive or deductive reasoning?

 

Terry used the distributive property and other known facts to create a series of new facts about the expression 3x3 + 6x2 + 3x. This is deductive reasoning

Answer

Deductive reasoning

 

 

Lucretia remembered this rule from arithmetic:

 

To tell if fractions  and  are equal, “cross multiply” to find ad and bc.
If ad = bc, then the fractions are equal.

 

 

She thought about whether this always works, and she made the following argument:

 

·         Say we know that ad = bc.

·         Divide both sides of the equation by bd—which we can do if b ≠ 0 and d ≠ 0—and simplify:

·        

 

·         So it’s true if b ≠ 0 and d ≠ 0.

·         If we’re starting with the fractions, then b and d can’t be 0, so this “cross-multiplication” rule works.

 

What kind of reasoning is Lucretia using?

 

A) Inductive reasoning

B) Deductive reasoning

 

Show/Hide Answer

A) Incorrect. When using inductive reasoning, you use specific examples and generalize them to a broader set of examples. In this case, Lucretia’s observations are already general, and she uses known facts to arrive at her final statement. If she were to use inductive reasoning, she might notice specific examples like  and 4 • 15 = 5 • 12, and  and 2 • 4 ≠ 3 • 3. From several examples like that, she could generalize that the equality of the fractions and the products found by cross-multiplication are the same for all pairs of fractions, and then she can make the conjecture that the fractions are equal if those products are equal.

 

B) Correct. Lucretia uses known facts to arrive at her final statement in a logical way, which is deductive reasoning.

 

 

Building a Logical Argument

 

The end result of deductive reasoning is often a logical argument. Most of the arguments we come across are polished, final arguments. We don't usually see the rough drafts, the mistakes, the lines of reasoning that went nowhere. We don't see that a writer started in a completely different place than the finished argument starts, and we can't tell when someone worked his argument out backwards because it was easier to figure it out from the conclusion than from the hypothesis!

 

The point? Don't worry when it's hard to find a good argument right away. Deductive reasoning often takes some creativity and persistence. Time and practice will make it easier to sort out the dead ends from the promising leads.

 

The example below provides one way to approach building an argument:

 

Example

Problem

Frank has three siblings, and they all like to chip in to buy lottery tickets. Whenever there's a big jackpot, he wonders if it could be evenly divided among the four of them, just in case they win. 

 

Frank has noticed that when a number formed by the last two digits of a whole number is divisible by 4, the whole number itself is divisible by 4. For example, 293,212 is divisible by 4 and 12 is divisible by 4.

 

How can he use deductive reasoning to justify the truth of this divisibility test?

Answer

First, let's write a general example of the conjecture. That is, this example has to represent many examples, not just one.

 

In this case, we want to talk about the last two digits of a whole number. Here’s a trick that can be helpful when working with digits: Use place values to write the number as a sum with addends that are products of powers of 10.

 

In this case, since Frank is interested in the last two digits of a larger number, we can separate those digits from the rest of the number. For example, 236 can be written as 200 + 36, or 2(100) + 36. The number 72,915 can be written as 72,900 + 15, or 729(100) + 15.

 

Any whole number can be written as 100a + b, where a and b are whole numbers and b < 100.

 

 

To put it in general terms, a large number can be written as 100 times a number plus a second number that is less than 100. This is a general, known statement.

 

 

 

If b is divisible by 4, then 100a + b is divisible by 4.

We can also write Frank's conjecture using algebraic expressions.

 

 

Now we need to find a way to connect the general example to the conclusion. This may take several steps.

 

In this case, we want 100a + b to be divisible by 4. So let’s factor the 4 out of the expression.

 

 

What does it mean for the number to be divisible by 4? It means the quotient is a whole number, so we want  to be a whole number. How can we show that it is?

 

Since a is a whole number, 25a is also a whole number.

We can use the hypothesis here. We defined a to be a whole number, so 25a has to also be a whole number.

 

Since b is divisible by 4,  is a whole number.

That means  is a whole number.

The hypothesis is that b is divisible by 4, so that gives us information about . Combine that with the information about 25a.

 

Since , we know that

. Since  is a whole number, 100a + b is divisible by 4.

Conclude the argument.

 

 

Here are some tips for building an argument. None of them will work every time, and we may need to try them all before we find an argument that works.

 

 

·         Use the hypothesis of the logical statement as a starting place.

 

·         Use the conclusion of the logical statement as a starting place.

 

·         Start with something related to either the hypothesis or conclusion that is already known to be true. For example, with an inequality using absolute value, perhaps start with the fact that |x| ≥ 0 for all real numbers x.

 

 

 

Build a logical argument to explain why a two-digit number is divisible by 3 if the sum of its digits are divisible by 3.

 

Hint: Write the two-digit number as 10x + y, and notice that 10x = 9x + x.

 

Show/Hide Answer

You may have a different argument than the one given here. That doesn’t mean your argument is wrong. However, you should read this argument to understand how it works. (You can learn a lot about building arguments by understanding how finished ones work.)

 

A two-digit number with tens digit x and ones digit y is equal to 10x + y. (For example, in 36, x = 3 and y = 6, so 36 = 10(3) + 6.)

 

The general statement of the conjecture is: “If x + y is divisible by 3, then 10x + y is divisible by 3.”

 

10x + y = 9x + x + y, so

If x + y is divisible by 3,  is a whole number. Since x is a digit, 3x is a whole number, so  is a whole number. That means the original number 10x + y is divisible by 3.

 

 

Summary

 

Deductive reasoning is a process of making conclusions by putting known facts together to provide a reasoned argument for a new fact. Deductive reasoning can be used to justify a conjecture arrived at through inductive reasoning. It's also helpful at times when inductive reasoning isn't appropriate, such as when there aren't enough examples to generalize from.