Section 2.3

1. Section 2.3 Deductive Reasoning

2. Symbolic Notation • Conditional Statements can be written using symbolic notation • p represents hypothesis • q represents conclusion • is read as “implies” • Therefore, a conditional statement looks like ______________ or ____________ If p, then q p q

3. q p If q, then p • To write the converse, we simply switch the p and q Symbolically, converse looks like • A biconditional statement can be written using symbolic notation as follows: • To write the inverse symbolically: ______________ • To write the contrapositive symbolically: __________ ______________ or ____________ p q p if and only if q _______________________ or _______________ ~p ~ q ~q ~ p

4. Example 1 Let p be “the value of x is -4” and q be “the square of x is 16”. a) Write in words b) Write in words c) Decide whether the biconditional statement is true. p q If x = -4, then x2 = 16 q p If x2 = 16, then x = -4 p q no

5. Example 2 Let p be “today is Monday” and q be “there is school” a) Write the contrapositive of symbolically and in words. a) Write the inverse of symbolically in words. p q ~q ~p If there is no school, then today isn’t Monday p q ~p ~q If today isn’t Monday, then there is no school.

6. Example 3 Let p be “a number is divisible by 3” and q be “ a number is divisible by 6” a) Write in words b) Write in words c) Is the biconditional statement true? d) Write the contrapositive of p q If a number is divisible by 3, then it is divisible by 6. q p If a number is divisible by 6, then it is divisible by 3. no p q If a number is not divisible by 6, then it is not divisible by 3.

7. Ex. 3 cont. p q e) Write the inverse of If a number is not divisible by 3, then it is not divisible by 6.

8. Types of Reasoning: • ____________________ uses facts, definitions, and accepted properties in a logical order to write a logical argument. • ___________________ uses previous examples and patterns to form a conjecture. Deductive Reasoning * Facts and definitions Inductive Reasoning * Past observations

9. Example 4 - 5 All dogs are mammals. All mammals have kidneys. Therefore all dogs have kidneys. All swans we have seen have been white; therefore all swans are white. Inductive or Deductive Inductive or Deductive

10. 2 Laws of Deductive Reasoning • ____________________ If p  q is a true conditional statement and p is true, then q is true. • ___________________ If pq and qr are true conditional statements, then pr is true. Law of detachment *If whole statement true, then both parts true Law of syllogism *look for statements linked together

11. Example 6 and 7 Michael knows that if he does not do his chores in the morning, he will not be allowed to play video games later the same day. Michael does not play video games on Friday afternoon. So Michael did not do this chores on Friday morning. If two angles are vertical, then they are congruent. <ABC and < DBE are vertical. So <ABC and <DBE are congruent. Law of syllogism doesn’t hold Valid or Invalid Valid or invalid Law of detachment

12. Example 8 Write some conditional statements that can be made from the following true statements using the Law of Syllogism. a) If a fish swims at 68 mi/h, then it swims at 110 km/h. b) If a fish can swim at 110 km/h, then it is a sailfish. c) If a fish is the largest species of fish, then it is a great white shark. d) If a fish weighs over 2000 lbs, then it is the largest species of fish. e) If a fish is the fastest species of fish, hten it can reach speeds of 68 mi/h. ____________________________________________________ _____________________________________________________ ________________________________________________________ If a fish swims 68 mi/h, then it is a sailfish. If a fish is the fastest species, then it is a sailfish. If a fish weighs over 2000 lbs, then it is a great white shark.