Deductive Reasoning

1. Deductive Reasoning

2. Objectives • I can identify an example of inductive reasoning. • I can give an example of inductive reasoning. • I can identify an example of deductive reasoning. • I can give an example of deductive reasoning. • I can write a conditional statement using symbolic notation.

3. Review

4. Symbolic Notation p : represents the hypothesis q : represents the conclusion → : is read as “implies” Ifthe sun is out, then the weather is good. p q The conditional statement can be written as: If p, then q or p → q

5. Symbolic Notation The converse of “If p, then q” is Ifthe weather is good, then the sun is out. q p The converse can be written as: If q, then p or q → p

6. Symbolic Notation A biconditional statement can be written as: If p, then q and if q, then p OR p↔ q OR pif and only if q

7. Using Symbolic Notation Let p be “the value of x is - 5” and let q be “the absolute value of x is 5” • Write p → q • Write q → p • Decide whether p↔ q is true. If the value of x is - 5, then the absolute value of x is 5. If the absolute value of x is 5, then the value of x is - 5. A) True. B) False, x can be 5. C) p↔ q is False.

8. Symbolic Notation The inverse of “If p, then q” is If, the sun is not out, thenthe weather is not good. ~ p ~ q The inverse can be written as: If ~ p, then ~q or ~ p→ ~ q

9. Symbolic Notation The contrapositive of “If p, then q” is Ifthe weather is not good, then the sun is not out. ~ q ~ p The contrapositive can be written as: If ~q, then ~ p or ~ q → ~ p

10. Using Symbolic Notation Let p be “it is raining” and let q be “the soccer game is canceled” • Write ~ p→ ~ q • Write ~ q → ~ p If it is not raining, then the soccer game is not canceled. If the soccer game is not canceled, then it is not raining.

11. Inductive vs. Deductive Reasoning • Inductive: previous examples and patterns are used to form a conjecture. • Andy knows that Kd is a sophomore and Maria is a junior. • All the other juniors that Andy knows are older than Kd. • Therefore, Andy reasons inductivelythat Maria is older than Kd based on past observations.

12. Inductive vs. Deductive Reasoning • Deductive: uses facts, definitions, and accepted properties in a logical order to write a logical argument. • Andy knows that Maria is older than Jason. • She also knows that Jason is older than Kd. • Andy reasons deductivelythat Maria is older than Kd based on accepted statements.

13. Laws of Deductive Reasoning • Law of Detachment: If p → q is true conditional statement and p is true, then q is true. Example: Devin knows that if he misses the practice the day before a game, then he will not be a starting player in the game. Devin misses practice on Tuesday so he concludes that he will not be able to start in the game on Wednesday.

14. Using the Law of Detachment • State whether the argument is valid. If two angles form a linear pair, then they are supplementary ; A and B are supplementary, so A and B form a linear pair. Not valid: p → q and q (the conclusion) are not true. The argument implies that all supplementary angles form a linear pair.

15. Laws of Deductive Reasoning • Law of Syllogism:If p → q and q → rare true conditional statements, then p → ris true. • p : Rebecka visits California. • q : Rececka spends New Year’s Day in Pasadena. • r : Rebecka goes to watch the Rose Parade. • If Rebecka visits California, then she will go to watch the Rose Parade.

16. Using the Law of Syllogism • Write some conditional statements that can be made from the following true statements using the Law of Syllogism. • If a bird is the fastest bird on land, then it is the largest of all birds. • If a bird is the largest of all birds, then it is an ostrich. • If a bird is the largest of all birds, then it is flightless.

17. Possible Conditional Statements • If a bird is the fastest bird on land, then it is an ostrich (1 & 2) • If a bird is the fastest bird on land, then it is flightless. (1 & 3)

18. Exit Card • Write the Contrapositive in symbolic notation. • Use the statements and Law of Syllogism to write a conditional statement both in words and symbols. • If a creature is a fly, then it has six legs. • If a creature has six legs, then it is an insect.