# Deductive reasoning

##### Intros
###### Lessons
1. Deductive Reasoning Overview:
2. Law of Detachment
3. Law of Syllogism
##### Examples
###### Lessons
1. Using the Law of Detachment
You are given the statements below. Assume these statements are true. Determine if a conclusion can be made by using the Law of Detachment.
1. If you are studying, then you are ready for the test.
You are ready for the test.
2. If you wake up 10 minutes earlier, then you will be on time for school.
You wake up 10 minutes earlier.
3. If you are hungry, then you will eat.
You will not eat.
2. Given the information and a statement below, determine if the conclusion is valid. Explain why it is valid or invalid.
If two lines are perpendicular, then they intersect
1. Given: the two lines are perpendicular.
Conclusion: the two lines intersect.
2. Given: the two lines intersect.
Conclusion: the two lines are perpendicular.
3. Using the Law of Syllogism
You are given the statements below. Assume these statements are true. Determine if a conclusion can be made by using the Law of Syllogism.
1. If the measure of an angle is 90°, then it is a right angle.
If it is a right angle, then it is not obtuse.
2. If you drive a car, then you can get to the grocery store in 30 minutes
If you get to the grocery store in 30 minutes, then you will get home on time
3. If you drink coffee, then you will stay awake
If you stay awake, then you drink coffee.
4. Given the information below, determine if the conclusion is valid. Explain why it is valid or invalid.
1. Given: If the shape is a square, then it is a regular polygon.
If it is a regular polygon, then all sides are equal.
Conclusion: If the shape is a square, then all sides are equal.
2. Given: If the shape is a square, then it is a regular polygon.
If it is a regular polygon, then all sides are equal.
Conclusion: If all sides are equal, then the shape is a square.
###### Topic Notes

Deductive Reasoning uses definitions, rules, and properties to see if a statement is true or false. There are two forms of deductive reasoning: Law of Detachment, and Law of Syllogism.

Law of Detachment states the following:

If $p\to q$ is true and $p$ is true, then $q$ must also be true

Law of Syllogism states the following:

If $p \to q$ and $q \to r$ are true, then $p \to r$ must be also true.