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4/8/2013 Definitions Logic is the study of the methods and principles of reasoning. An argum ent uses a set of facts or assumptions, called MAT 110 prem ises, to support a conclusion . Chapter 1 Notes Premises are facts


  1. 4/8/2013 Definitions • Logic is the study of the methods and principles of reasoning. • An argum ent uses a set of facts or assumptions, called MAT 110 prem ises, to support a conclusion . Chapter 1 Notes ▫ Premises are facts ▫ Conclusions are supported by premises Logic & Reasoning David J. Gisch Fallacy S tructures Definitions • Logic is the study of the methods and principles of Appeal to Popularity Appeal to Em otion reasoning. Logic False Cause Personal Attack • An argum ent uses a set of facts or assumptions, called Appeal to Ignorance Circular Reasoning prem ises, to support a conclusion . Argument Hasty Generalization Diversion -Red Herring • A fallacy is a deceptive argument—an argument in which the conclusion is not well supported by the Lim ited Choice Straw Man premises. Fallacy 1

  2. 4/8/2013 Definitions • A proposition makes a claim (either an assertion or a denial) that may be either true or false. It must have the structure of a complete sentence. • Any proposition has two possible truth values : T = true or F = false. Propositions and Truth Values • A truth table is a table with a row for each possible set of truth values for the propositions being considered. Propositions and Truth Values Are they propositions? • It is a proposition if: • Joan is sitting in a chair. ▫ It is a complete sentence ▫ It makes a claim • I did not take the pen. ▫ The claim can be true or false • It will not be a proposition if: • Are you going to the store? ▫ It is a question ▫ Does not assert or deny anything • Three miles south of here ▫ Is not a complete sentence • 7 + 9 = 2 2

  3. 4/8/2013 Negation (Opposites) What would the negative be? The negation of a proposition p is another proposition • Joan is sitting in a chair. that makes the opposite claim of p . • I took the pen. p not p ← If p is true (T), not p is false (F). T F • Betsy is the fastest runner on the team. ← If p is false (F), not p is true (T). F T • 7 + 9 = 2 Symbol: ~ S etting up a Truth Table Double Negation The double negation of a proposition p , not not p , • The number of row depends on the number of has the same truth value as p . combinations. ▫ If you have two statements, each statement can be true or false so that is 2 � 2 � 4 combinations or rows. p not p not not p p q p and q Each row represents a possible T F T T T T combination of p and q. T F F F T F F T F F F F ▫ If you have three statements, each statement can be true or false so that is 2 � 2 � 2 � 8 combinations or rows. 3

  4. 4/8/2013 Double Negations (Opposites) Logical Connectors Propositions are often joined with logical • Double negation has the same truth value as the original connectors —words such as and , or , and if…then . proposition • It’s like turning over a coin Example: ▫ Start with heads p = I won the game. ▫ Turn it over once, tails q = It was fun. ▫ Turn it over again, back to heads Logical Connector New Proposition and I won the game and it was fun. or I won the game or it was fun. if…then If I won the game, then it was fun. And S tatements (Conj unctions) NOTE Given two propositions p and q , the statement p and q •A conjunction is only true if both is called their conjunction . It is true only if p and q are p and q are true. both true. p q p and q p q p and q T T T T T T T F F T F F F T F F T F F F F F F F Symbol: 4

  5. 4/8/2013 Or S tatements (Disj unctions) Or S tatement ( Disj unctions) The word or can be interpreted in two distinct ways: • Example: INCLUSION • A health insurance policy covers hospitalization in cases of illness or injury. • An inclusive or means “either or both.” ▫ Covers illness Or ▫ Covers injury Or • An exclusive or means “one or the other, but not both.” ▫ Both In logic, assume or is inclusive unless told otherwise. Or S tatements (Disj unctions) Or S tatement (Disj unction) Given two propositions p and q , the statement p or q • Example:EXCLUSION is called their disjunction . It is true unless p and q are both false. • A restaurant offers soup or salad. ▫ Offers soup p q p or q Or ▫ Offers salad T T T NOT ▫ Both T F T F T T F F F Symbol: 5

  6. 4/8/2013 NOTE If … Then S tatement (Conditional) • If all politicians are liars then Representative Smith is a •A disjunction is true unless both p liar. and q are false. • Conditional propositions ▫ p is called the p q p or q  Hypothesis or  Antecedent T T T ▫ q is called the  Conclusion T F T  Consequence F T T ▫ q is true on the condition that p is true. F F F If… Then S tatements (Conditionals) If… Then A statement of the form if p, then q is called a Think of If-Then statements as a rule. conditional proposition (or implication). It is true unless p is true and q is false. RULE: If your grade is greater than 94%, then you get an A. p q if p, then q 1. Sally gets a 98%, and therefore an A. T T T 2. Sally got a 98%, but got a B. T F F F T T 3. Sally got a 90%, and received an A. F F T 4. Sally got a 60% and received a C.  Proposition p is called the hypothesis .  Proposition q is called the conclusion . 6

  7. 4/8/2013 Truth Tables Truth Tables � � ∼ � � ∨∼ � � � �⋀� � ∨ � Truth Table Practice Truth Table Practice p  q ~ p  ~ q ~ ( p  q ) p  q ~ p  ~ q ~ ( p  q ) ~ p ~ q ~ p ~ q p q p q T T T T F F T F F T F T F F T F T F F F F Practice by writing the Practice by writing the Note: ~ signifies NEGATION Note: ~ signifies NEGATION  signifies AND truth values of each row  signifies AND truth values of each row  signifies OR in the table above.  signifies OR in the table above. 7

  8. 4/8/2013 Truth Table Practice Truth Table Practice p  q ~ p  ~ q ~ ( p  q ) p  q ~ p  ~ q ~ ( p  q ) ~ p ~ q ~ p ~ q p q p q T T F F T F F T T F F T F F T F F T F T T T F F T F T T F T F T T F F T T F F F F Practice by writing the Practice by writing the Note: ~ signifies NEGATION Note: ~ signifies NEGATION  signifies AND truth values of each row  signifies AND truth values of each row  signifies OR in the table above.  signifies OR in the table above. Truth Table Practice p  q ~ p  ~ q ~ ( p  q ) ~ p ~ q p q T T F F T F F T F F T F T T Sets and Venn Diagrams F T T F F T T F F T T F T T Practice by writing the Note: ~ signifies NEGATION  signifies AND truth values of each row  signifies OR in the table above. 8

  9. 4/8/2013 Definition Ellipses • A set is a well-defined collection of objects. • Use three dots, …, to indicate a continuing pattern if ▫ We denote sets with capital letters there are too many members to list. For example, ▫ We write sets with brackets as follows 3, 4, 5 ▫ This is referred to as roster form of a set. {1945, 1946, 1947 . . . 1991} • Any item belonging to a set is called an elem ent or You need to list three items to establish a clear m em ber of that set. { 6, 7, 8 . . .} pattern! ▫ We denote elements of a set as follows 3 ∈ 3, 4, 5 { . . ., -3, -2, 1, 0, 1, 2, . . .} 7 ∉ 3, 4, 5 Why well-defined? Give me the set of people in this room who are nice. Definition Definition • Repetitions of elements do not matter. Whether it is • The set of all things being discussed is referred to as the listed once or twice it is still a member of the set and that universal set . We denote the universal set as set � . is all that matters. • Order also does not matter in sets, unless it is used to • For example, if we were discussing arithmetic in third establish a pattern. grade we might use the universal set of whole numbers. 3, 4, 5 � 4, 3, 5 � 3, 3, 3, 4, 5 � 5, 5, 3, 4, 4, 4 In college algebra the universal set would be all real numbers. 9

  10. 4/8/2013 The Real Numbers S ets Example: Let the universal set be the set of real numbers. Example: Write out each of the following sets in roster form. Natural = { (a) The set of all numbers (integers) between 2 and 7. Whole = { Integers = { (b) The set days of the week that begin with the letter S. Rational = { Irrational = { (c) The set of planets in our solar system that begin with the letter C. S ets and Propositions S ets and Propositions • There are four standard categorical • Now that we’ve been introduced to sets and have studied a little bit about set we are ready to discuss propositions propositions that make claims about sets. – All S ubject are P redicate – No S ubject are P redicate • As you know, Propositions are in the form of complete – Some S ubjec t are P redicate sentences. • The sets referenced in a proposition can be identified as – Some S ubject are not P redicate follows: – Note : ▫ one set appears in the subject of the sentence • S (propositions in the subject) ▫ one set appears in the predicate of the sentence. • P (propositions in the predicate) 10

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