Section 1 1 Statements Reasoning 2102022 1 Statement

Section 1. 1 Statements & Reasoning 2/10/2022 1

Statement • Group of words & symbols, classified collectively as true or false, simple or compound. Questions, commands are not statements! Why? – They cannot be determined to be true or false • Simple: Snow is cold. • Compound: made up of two or more simple statements, connected by ‘and, ’ ‘or, ’ ‘if…then’ 2/10/2022 2

Negation • ‘Not’ • Changes the truth value of the statement to its opposite. • Let P be the statement “My car is white. ” • Then the negation of P is ‘not P’ or ~P. • “My car is not white. ” • What is the negation of “Some”? – Some men have beards – Not one man has a beard. 2/10/2022 3

Conjunction • Compound statement • Signified by ‘and’ • Ex. Snow is cold and rain is wet. • True only if both are? True. 2/10/2022 4

Disjunction • Compound statement • ‘Or’ • You can have ice cream or strawberries for dessert. • False only if both are ? False 2/10/2022 5

Conditional statement • Also called implication • ‘If…then, ’ symbolized by using • Ex. Let P represent the statement, “The sun is shining, ” and Q represent the statement, “I can see my shadow, ” then the conditional statement, “If the sun is shining, then I can see my shadow. ” • Symbolized by P Q 2/10/2022 6

Conditional (cont’d) • P →Q • P is the hypothesis and Q is the conclusion. 2/10/2022 7

3 types of Reasoning: 1. Intuition: An idea leading to a statement of a theory. – You enter the bank and the line is very long. You conclude that you will have a long wait. – Before the opening kickoff of the first game of the season, Bill predicts his team will win. Examples 5 p. 4 2/10/2022 8

2. Inductive reasoning Induction: Using specific observations to draw a general conclusion (from specific to general). – You find a bag of tennis balls and the first 3 are flat. You conclude that the whole bag is flat. – After examining and diagnosing several patients, the doctor concludes that there is a flu epidemic in that area. • Involves examining a few examples, observing a pattern, and then assuming that the pattern will never end. • Not a valid proof, although it often suggests statements that can be proved by other methods. 2/10/2022 9

Example of Inductive Reasoning String of odd integers 1+3+5+7+9 Sum 4 9 16 25 Do you notice a pattern? What conclusion can you draw? Ex 6 -7 p. 5 2/10/2022 10

3. Deductive reasoning Deduction: Accept that certain assumptions are true which guarantee a specific conclusion. – You know that the movie is two hours and starts at 8 pm. You conclude that it will end at 10 pm. – If an integer is even, then it is divisible by two. Since 14 is an even integer, it is divisible by two. • May be considered the opposite of inductive reasoning. • Uses accepted facts, i. e. undefined terms, postulates, & previously established theorems, to reason in a step-by-step fashion until a desired conclusion is reached. 2/10/2022 11

An easy example of Deduction Assume the following 2 postulates are true: 1) All last names that have 7 letters with no vowels are the names of Martians. 2) All Martians are 3 feet tall. Prove that Mr. Xhzftlr is 3 feet tall. 2/10/2022 12

Proof • Use the 2 -column format: Statements 1. The name is Mr. Xhzftlr. 2. Mr. Xhzftlr is a Martian. Reasons 1. Given. 2. All last names with no vowels are the names of Martians (Post. 1) 3. Mr. Xhzftlr is 3 feet tall. 3. All Martians are 3 feet tall (Post. 2). Notice that each statement has a corresponding justification. Ex. 8 p. 6 2/10/2022 13

Law of Detachment A form of deductive reasoning If statements 1 and 2 (premises) are true, then the conclusion is true. 1. If P, then Q premises 2. P. Q conclusion • P: It is raining • Q: The field is wet • If P Q • If it is raining, then the field will be wet. CANNOT MAKE ANY STATEMENT CONCERNING “IF Q THEN P. ” (Fallacy of the Converse) • If it is raining then the field will be wet • If the field is wet, does that mean it is raining? • Give other options • Examples 9 – 10 p. 6 - 7 2/10/2022 14