**Deductive Reasoning :**

This section will hopefully be a refresher topic on deductive reasoning. Of course, we cannot let you get away that easy!

Now that you have a good understanding of deductive reasoning, we will add some more applications for this concept. You will learn about the Law of Syllogism, which you may have also seen in a previous math class.

You would have learned it as the Transitive property. Don’t worry, even if you don’t remember that property, we will show you how it works.

**What is deductive reasoning ? **

**Answer : **

The process of drawing logically certain conclusions by using an argument is known as deductive reasoning.

Deductive reasoning is a valid form of proof.

**Law of Detachment : **

An if-then statement is a form of deductive reasoning.

We assume that if the "if" part is true, then, by the Law of Detachment, it automatically follows that the "then" part is always true.

**Example :**

If you take this medicine regularly, you will be recovered soon.

**Law of Syllogism : **

Law of Syllogism is nothing but the transitive property of logical statements.

That is,

If p → q and q → r, then p → r

**Example : **

Two if-then statements :

1. If an object is a square, then it is a rhombus.

2. If an object is a rhombus, then it is an equilateral.

By Law of Syllogism, we have

That is,

If an object is a square, then it is an equilateral

**Example 1 : **

Draw a logical conclusion from the given statements :

I can go to the party, if I clean my room.

I cannot go to the party.

**Answer : **

By Law of Detachment, we can have the following conclusion for the above statements.

I did not clean my room

**Example 2 : **

What conclusion can be drawn :

If I do my homework, then I will get good grades.

If I get good grades, then my parents will be happy.

**Answer : **

By Law of Syllogism, we can have the following conclusion for the above statements.

If I do my homework, then my parents will be happy.

**Example 3 :**

Write some conditional statements that can be made from the following true statements using the Law of Syllogism.

1. If a bird is the fastest bird on land, then it is the largest of all birds.

2. If a bird is the largest of all birds, then it is an ostrich.

3. If a bird is a bee hummingbird, then it is the smallest of all birds.

4. If a bird is the largest of all birds, then it is flightless.

5. If a bird is the smallest bird, then it has a nest the size of a walnut half-shell.

**Answer :**

Here are the conditional statements that use the Law of Syllogism.

a. If a bird is the fastest bird on land, then it is an ostrich. (Use 1 and 2.)

b. If a bird is a bee hummingbird, then it has a nest the size of a walnut half-shell. (Use 3 and 5.)

c. If a bird is the fastest bird on land, then it is flightless. (Use 1 and 4.)

After having gone through the stuff above, we hope that the students would have understood deductive reasoning.

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