Deductive Reasoning

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.

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

$2n\times (2m+1)$

Then we can expand through to get,

$2mn+2n$

And factor out the 2 to get,

$2(mn+n)$

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

repeated to infinity.

Solution

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

$1-A=1-(1-1+1-1+1-1...)$

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!

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.

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.

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 $x.$