2-4: Deductive Reasoning
Objectives:
1. To define Deductive Reasoning
2. To use the Law of Detachment
3. To use the Law of Syllogism
Deductive Reasoning
:
The process of reasoning logically from given
statements to a conclusion.
Remember:
Inductive reasoning uses pattern recognition to
come to a conclusion that may or may not be
true.
Deductive reasoning uses rules to come to a
conclusion that is certainly true.
The Parts of Deductive Reasoning
• An undefined term is a word that we describe but don’t
formally define (remember point, line, and plane).
• A definition uses known words to describe a new word.
• A postulate is a statement that we assume to be universally
true.
• A theorem is a statement that has been proven true using
definitions, postulates, or other theorems.
Law of Detachment
If a conditional and its hypothesis are both true,
then the conclusion must also be true.
If we’re told (given) that the following are true:
• 𝑝 → 𝑞 “The statement ‘if p then q’ is true.”
and
• 𝑝
“p is true.”
then the Law of Detachment says that we can
conclude:
• 𝑞
“q must be true.”
Using Law of Detachment
What can you conclude from the given statements?
1) 𝑝 → 𝑞: If a person sees lightning nearby, then it is
not safe for that person to be out in the open.
𝑝: Marla sees lightning nearby from the soccer
field.
𝑞: It is not safe for Marla to be out in the open.
2) 𝑝 → 𝑞: If an angle has a measure of 90°, then it is
a right angle.
𝑝: Angle A has a measure of 90°.
𝑞: Angle A is a right angle.
Law of Syllogism
Allows you to state a conclusion from two true
conditional statements when the conclusion of the first
statement is the hypothesis of the second statement.
Symbolic Form:
If 𝑝 → 𝑞 and 𝑞 → 𝑟 are true statements, then 𝑝 → 𝑟 is a true
statement.
Example:
If a number is prime, then it does not have repeated factors.
If a number does not have repeated factors, then it is not a
perfect square.
Conclusion: If a number is prime, then it is not a perfect square.
Using the Law of Syllogism
If a number ends in 0, then its divisible by 10.
If a number is divisible by 10, then it is divisible by
5.
Conclusion: If a number ends in 0, then it is divisible
by 5.
If a number ends in 6, then its divisible by 2.
If a number ends in 4, then its divisible by 2.
Conclusion: We can’t make any new conclusion!
Using Both Laws
If a river is more than 4000 mi. long, then it is
longer than the Amazon.
If a river is longer than the Amazon, then it is
the longest river in the world.
The Nile is 4132 mi. long.
What can you conclude?
Do the 2-4 assignment in MathXL