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Section 1.1

Section 1.1. Statements & Reasoning. 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.

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Section 1.1

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  1. Section 1.1 Statements & Reasoning

  2. 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’

  3. 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.

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

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

  6. 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

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

  8. 3 types of Reasoning:1. Intuition 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

  9. 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.

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

  11. 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, defined terms, postulates, & previously established theorems, to reason in a step-by-step fashion until a desired conclusion is reached.

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

  13. Proof • Use the 2-column format: StatementsReasons 1. The name is Mr. Xhzftlr. 1. Given. 2. Mr. Xhzftlr is a Martian. 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

  14. Law of DetachmentA form of deductive reasoningIf 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

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