# Deductive Reasoning

If you go to buy a car, you know that that car is going to have wheels. Why? Because intuitively you know that since all cars have wheels, the one you wish to buy will too.

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How about when you go to a bookstore to buy a physical book, you will always know that that book will have pages. Why? Because intuitively you know that since all physical books have pages, the one you are going to buy will too.

These are examples of how we use deductive reasoning in our lives every day without even realizing it. Not only that, but in a large number of math questions that you have ever answered, you have used deductive reasoning.

## Deductive reasoning Definition

Deductive reasoning is the drawing of a true conclusion from a set of premises via logically valid steps. A conclusion can be said to be deductively valid if both conclusion and premises are true.

This may seem a tricky concept to grasp at first due to the novel terminology, but it really is quite simple! Any time that you work out an answer with certainty from some initial information, you have used deductive reasoning.

Deductive reasoning really can be understood as drawing facts from other facts, and in essence, is the process of drawing specific conclusions from general premises.

Facts → Facts

General Premises → Specific Conclusions

Let's take a look at some examples of deductive reasoning to make this clearer.

## Deductive reasoning examples

Jenny is told to solve the equation $2x+4=8$, she uses the following steps,

$2x+4-4=8-4$

$2x=8$

$2x÷2=8÷2$

$x=4$

As Jenny has drawn a true conclusion, $x=4$, from the initial premise, $2x+4=8,$ this is an example of deductive reasoning.

Bobby is asked the question ' x is an even number less than 10, not a multiple of 4, and not a multiple of 3. What number is x?' As it must be an even number less then $10$, Bobby deduces that it must be $2,4,6,$ or $8.$ As It is not a multiple of $4$ or $3$ Bobby deduces it cannot be $4,6,$ or $8.$ He decides, therefore, it must be $2.$

Bobby has drawn a true conclusion, $x=2,$ from the initial premises that $x$ is an even number less than $10$that is not a multiple of $4$ or $3.$ Therefore, this is an example of deductive reasoning.

Jessica is told all angles less than $90°$ are acute angles, and also that angle $A$ is $45°.$She is then asked if angle $A$ is an acute angle. Jessica answers that since angle $A$ is less than $90°,$ it must be an acute angle.

Jessica has drawn a true conclusion that angle $A$ is an acute angle, from the initial premise that all angles less than $90°$ are acute angles. Therefore, this is an example of deductive reasoning.

Not only are these all examples of deductive reasoning, but did you notice we have used deductive reasoning to conclude that they are in fact examples of deductive reasoning. That's enough to make anyone's head hurt!

Some more everyday examples of deductive reasoning might be:

• All tuna have gills, this animal is a tuna - therefore it has gills.
• All brushes have handles, this tool is a brush - therefore it has a handle.
• Thanksgiving is on the 24th of November, today is the 24th of November - therefore today is thanksgiving.

On the other hand, sometimes things that may appear to be sound deductive reasoning, in fact, are not.

## Method of deductive reasoning

Hopefully, you are now familiar with just what deductive reasoning is, but you might be wondering just how you can apply it to different situations.

Well, it would be impossible to cover how to use deductive reasoning in every single possible situation, there are literally infinite! However, it is possible to break it down into a few key tenets that apply to all situations in which deductive reasoning is employed.

In deductive reasoning, it all starts with a premise or set of premises. These premises are simply statements that are known or assumed to be true, from which we can draw a conclusion through the deductive process. A premise could be as simple as an equation, such as $5{x}^{2} +4y=z,$ or a general statement, such as 'all cars have wheels.'

Premises are statements that are known or assumed to be true. They can be thought of as starting points for deductive reasoning.

From this premise or premises, we require to draw a conclusion. To do this, we simply take steps toward an answer. The important thing to remember about deductive reasoning is that every step must follow logically.

For instance, all cars have wheels, but that does not mean that logically we can assume anything with wheels is a car. This is a leap in logic and has no place in deductive reasoning.

If we were asked to determine the value of $y$ from the premises,

$5{x}^{2}+4y=z$, $x=3,$and $z=2,$

then the logical steps we could take to draw a conclusion about the value of $y$ might look like this,

Step 1. Substituting the known values of $x$ and $z$ yields $5×{3}^{2}+4y=2$

Step 2. Simplifying the expression yields $45+4y=2$

Step 3. Subtracting 45 from both sides yields $4y=-43$

Step 4. Dividing both sides by 4 yields $y=-10.75$

We can check in this instance that the conclusion we have drawn is in-line with our initial premises by substituting the obtained value of y, as well as the given values of x and z into the equation to see if it holds true.

$5{x}^{2}+4y=z$

$5×{3}^{2}+4×\left(-10.75\right)=2$

$45-43=2$

$2=2$

The equation does hold true! Therefore we know that our conclusion is in-line with our three initial premises.

You can see that each step to reach the conclusion is valid and logical.

For instance, we know in step 3 that if we subtract 45 from both sides, both sides of our equation will remain equal, ensuring that the yielded expression is a true fact. This is a fundamental tenet of deductive reasoning, a step taken to draw a conclusion is valid and logical so long as the statement or expression obtained from it is a true fact.

## Solving deductive reasoning questions

Let's take a look at some questions that might come up regarding deductive reasoning.

Stan is told that every year for the last five years, the population of grey squirrels in a forest has doubled. At the start of the first year, there were $40$ grey squirrels in the forest. He is then asked to estimate how many rabbits there will be $2$ years from now.

Stan answers that if the trend of the population doubling every two years continues then the population will be at $5120$ in $2$ years time.

Did Stan use deductive reasoning to reach his answer?

Solution

Stan did not use deductive reasoning to reach this answer.

The first hint is the use of the word estimate in the question. When using deductive reasoning, we look to reach definite answers from definite premises. From the information given, it was impossible for Stan to work out a definite answer, all he could do was make a good attempt at a guess by assuming that the trend would continue. Remember, we are not allowed to make assumptions in our steps when using deductive reasoning.

Prove with deductive reasoning that the product of an odd and even number is always even.

Solution

We know that even numbers are integers that are divisible by $2$, in other words $2$ is a factor. Therefore we can say that even numbers are of the form $2n$ where $n$ is any integer.

Similarly, we can say that any odd number is some even number plus $1$ so we can say that odd numbers are of the form $2m+1$, where $m$ is any integer.

The product of any odd and even number therefore can be expressed as

$2n×\left(2m+1\right)$

Then we can expand through to get,

$2mn+2n$

And factor out the 2 to get,

$2\left(mn+n\right)$

Now, how does this prove that the product of an odd and even number is always even? Well, let's take a closer look at the elements inside the brackets.

We already said that $n$ and $m$ were just integers. So, the product of $m$ and $n$, that is $mn$ is also just an integer. What happens if we add two integers, $mn+n$, together? We get an integer! Therefore our final answer is of the even number form we introduced at the beginning, $2n$.

We have used deductive reasoning in this proof, as in each step we have used sound logic and made no assumptions or leaps in logic.

Find, using deductive reasoning, the value of A, where

$A=1-1+1-1+1-1+1...$

repeated to infinity.

Solution

One way to solve this, is to first take $A$ away from one.

$1-A=1-\left(1-1+1-1+1-1...\right)$

Then, by expanding the brackets on the right-hand side we get,

$1-A=1-1+1-1+1-1+1...$

1 - A = 1 - 1 -1+ 1 - 1 + 1 -1...

Hmmm, does that right-hand side seem familiar? It's just $A$ of course! Therefore

$1-A=A$

Which we can simplify to

$2A=1$

$A=\frac{1}{2}$

Hmmm, that's odd! It's not an answer that you would expect. In fact, this particular series is known as Grandi's Series, and there is some debate amongst mathematicians over whether the answer is 1, 0, or 1/2. This proof however is a good example of how deductive reasoning can be used in math to seemingly prove strange and unintuitive concepts, sometimes it's just about thinking outside of the box!

## Types of deductive reasoning

There are three primary types of deductive reasoning, each with its own fancy-sounding name, but really they are quite simple!

### Syllogism

If $A=B$ and $B=C,$ then $A=C.$ This is the essence of any syllogism. A syllogism connects two separate statements and connects them together.

For instance, if Jamie and Sally are the same age, and Sally and Fiona are the same age, then Jamie and Fiona are the same age.

An important example of where this is used is in thermodynamics. The zeroth law of thermodynamics states that if two thermodynamic systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.

### Modus Ponens

A implies B, since A is true then B is also true. This is a slightly complicated way of terming the simple concept of modus ponens.

An example of a modus ponens could be, all shows on a tv channel are less than forty minutes long, you are watching a show on that tv channel, therefore the show you are watching is less than forty minutes long.

A modus ponens affirms a conditional statement. Take the previous example. The conditional statement implied in the example is 'if the show is on this tv channel, then it is less than forty minutes long.'

### Modus Tollens

Modus tollens are similar, but opposite to modus ponens. Where modus ponens affirm a certain statement, modus ponens refute it.

For instance, in Summer the sun sets no earlier than 10 o'clock, today the sun is setting at 8 o'clock, therefore it is not Summer.

Notice how modus tollens are used to make deductions that disprove or discount something. In the example above, we have used deductive reasoning in the form of a modus tollens not to deduce what season it is, but rather what season it is not.

### Types of Deductive Reasoning Examples

Which type of deductive reasoning has been used in the following examples?

(a)${x}^{2}+4x+12=50$ and ${y}^{2}+7y+3=50$, therefore ${x}^{2}+4x+12={y}^{2}+7y+3$.

(b) All even numbers are divisible by two, $x$ is divisible by two - therefore $x$ is an even number.

(c) All planes have wings, the vehicle I am on does not have wings - therefore I am not on a plane.

(d) All prime numbers are odd, 72 is not an odd number, 72 cannot be a prime number.

(e) Room A and Room B are at the same temperatures, and Room C is the same temperature as Room B - therefore Room C is also the same temperature as Room A

(f) All fish can breathe underwater, a seal cannot breathe underwater, therefore it is not a fish.

Solution

(a) Syllogism - as this deductive reasoning is of the form $A=B,$ and $B=C$, therefore $A=C.$

(b) Modus Ponens - as this deductive reasoning is affirming something about $x.$

(c) Modus Tollens - as this deductive reasoning is refuting something about $x.$

(d) Modus Tollens - once again this deductive reasoning is refuting something about $x.$

(e) Syllogism - this deductive reasoning is also of the form $A=B$ and $B=C,$ therefore $A=C.$

(f) Modus Ponens - this deductive reasoning is affirming something about $x.$

## Deductive Reasoning - Key takeaways

• Deductive reasoning is a type of reasoning that draws true conclusions from equally true premises.
• In deductive reasoning, logical steps are taken from premise to conclusion, with no assumptions or leaps in logic made.
• If a conclusion has been reached using flawed logic or assumption then invalid deductive reasoning has been used, and the conclusion drawn cannot be considered true with certainty.
• There are three types of deductive reasoning: syllogism, modus ponens, and modus tollens.

#### Flashcards in Deductive Reasoning 12

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What is deductive reasoning in math?

Deductive reasoning is a type of reasoning that draws true conclusions from equally true premises.

What is an advantage of using deductive reasoning?

Conclusions drawn using deductive reasoning are true facts, whereas conclusions drawn with inductive reasoning may not necessarily be true.

What is deductive reasoning in geometry?

Deductive reasoning can be used in geometry to prove geometric truths such as the angles in a triangle always add up to 180 degrees.

What is the difference between deductive and inductive reasoning?

Deductive reasoning produces specific true conclusions from true premises, whereas inductive reasoning produces conclusions that seem as if they could logically be true, but aren't necessarily, from specific premises.

How are deductive and inductive reasoning similar?

Deductive and inductive reasoning are both used to draw conclusions from a set of premises.

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