Deductive Reasoning and Paragraph Proofs

1. Deductive Reasoning and Paragraph Proofs 2.4 Ms. Verdino

2. Biconditional Statements • Biconditional Statement: use this symbol ↔ • Example • Biconditional Statement: The weather is good if and only if the sun is out • p: the sun is out • q: the weather is good • P if and only if q, or q ↔ p

3. Review Of Symbolic Logic • Conditional Statement • p → q • Converse • q → p • Inverse • ~p → ~q • Contrapositive • ~q → ~p • Biconditional • p  q

4. Using the laws of logic • Inductive Reasoning • Looking at several specific situations to arrive at a conjecture • Deductive Reasoning • Uses a rule to make a conjecture

5. Logical Argument • Using facts, definitions, and accepted properties in a logical order • Laws of Deductive Reasoning • Law of Detachment • Law of Syllogism

6. Law Of Detachment • If p → q is a true statement and p is true, then q is true

7. Using the Law of Detachment • Given: If a student gets an A on her final then she will pass the course. Felicia got an A on her history final. • What can we conclude using the law of detachment?

8. Using the Law of Detachment • Given: If there is lightning, then it is not safe to be out in the open. Maria sees lightning from the soccer field • If a figure is a square, then its sides have equal length. Figure ABCD has sides of equal length. • What can we conclude using the law of detachment?

9. Law Of Syllogism • If p → q and q → r are true conditionals, then p → r is also true

10. Using the Law Of Syllogism • Given: If a figure is a square, then the figure is a rectangle. If the figure is a rectangle, then the figure has four sides. • What can we conclude using the law of syllogism?

11. Using the Law of Syllogism • Given: If a whole number ends in 0, then it is divisible by 10. If a whole number is divisible by 10, then it is divisible by 5. • What can we conclude using the law of syllogism?

12. Proof • Proof: a logical argument in which each step towards the conclusion is supported by a definition, postulate, or theorem i.e. supporting statements. • Paragraph Proof: a type of proof that gives the steps and supporting statements in paragraph form

13. Writing a proof • Steps to writing a good proof: • List the given information. • State what is to be proven. • Draw a diagram of the given information. • Develop an argument that properly uses deductive reasoning. • Each step follows from the previous • Each step is supported by accepted facts.

14. Paragraph Proof Given: M is the mid point of PQ • Prove: PM = MQ • Proof: From the definition of midpoint of a segment, PM = MQ. i.e. the measures of the segments are equal. This means that PM and MQ have the same measure. By definition of congruence, if two segments have the same measure, then they are congruent. Thus, PM = MQ

15. Paragraph Proof • In the figure at the right, P is the midpoint of QR and ST and QR = ST • Write a paragraph proof to show that PQ = PT • Given: • P is the midpoint of QR and ST • And QR = ST • Prove: PQ = PT • From the definition of midpoint of a segment, QP = PR. From the segment addition postulate, QR = QP + PR. Substituting QP for PR, QR = QP + QP. Simplifying gives QR = 2QP. Likewise, ST = 2PT. From the definition of congruent segments, QR = ST from the given information. Substituting 2QP for QR and 2PT for ST gives 2QP = 2PT. Dividing both sides by 2 gives QP = PT. Since QP is another way of naming the segment PQ, we can rewrite the last equation as PQ = PT and reach our desired conclusion.