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
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
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
4/8/2013 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
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
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
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
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
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
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
Recommend
More recommend
