1. Essential Question: What is deductive reasoning?

2. →: implies ↔: if and only if Symbolic Notation p: hypothesis
q: conclusion
→: implies
∼ : negation
↔: if and only if
p→q: If p, then q.

3. Examples: Symbolic Notation
Let p be “the value of x is 3” and let q be “x³=27.”
1. Write p → q in words.
2. Write q → p in words.
3. Write p ↔ q in words.

4. Examples: Symbolic Notation
Let p be “ a word is misspelled” and q be “a word is not in the dictionary.”
1. Write the contrapositive of p→q.
2. Write the inverse of p→q.

5. Deductive Reasoning
Deductive reasoning is a type of reasoning in which facts are used to make a conjecture.
Two types of deductive reasoning:
Law of Detachment
Law of Syllogism

6. Example; Inductive or deductive reasoning?
Julie is researching computers. She knows that brand X is cheaper than brand Y. She also knows that Brand Y is cheaper than Brand Z. Therefore, brand X is the cheapest of all computers.

7. Jay knows that Frank is older than David
Jay knows that Frank is older than David. He also knows that David is older than Javier. Therefore, Jay knows that Frank is older than Javier.

8. Law of Detachment
If p→q is a true conditional statement and p (the hypothesis) is true, then q (the conclusion) is true.
Law of detachment only has one if-then sentence (you may have to rewrite the first sentence in if-then form)
If the conclusion from the first sentence is the second sentence, then it is no conclusion

9. Law of Syllogism
If p→q and q→r is a true conditional, then p→r is true.
Law of Syllogism will have two if-then statements
The conclusion of the first sentence MUST BE THE SAME as the hypothesis of the second sentence.

10. If an angle is acute, then it is not obtuse. ∠ABC is acute.
Examples: Determine if a valid conclusion can be made from the following statements. If so, state it and the type of reasoning used.
If an angle is acute, then it is not
obtuse.
∠ABC is acute.

11. Right angles are congruent.
∠A≅∠B
3. If you save a penny, then you have earned a penny.
Art saves a penny.

12. If you are a teenager, then you are always right.
If you are always right, then people will listen to you.
If you drive 50 mph in a school zone, then you will get a speeding ticket.
Pat received a speeding ticket.