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Logic Unit L.0 Quadratic Formula, Painted Cube Activity Worksheet D#1-17 ax^2+bx+c = 0 b b 2 4ac x 2a - work in teams on 3 pages related to cubes - homework is also to finish what you don’t get done in class L.1 Introduction to Logic Part 1 exercises (all) - Logic – the study of arguments - we will learn how to analyze existing statements in a rational way and be able to deduce new ones - you might have seen this on Star Trek, Sherlock Holmes, debate, famous philosophers like Plato and Confucious,…. - simple statement – a statement that is either T or F and contains no other statement inside it - can assign a variable to a statement: c - “I like the Colts” ~c means “I don’t like them - V = means OR (disjunction) - ^ (upside down V) means AND (conjunction) - can form complex statements with AND/OR - (p V q) is true if one of them is true - (p ^ q) is true if both are true - do exs where are given p,q as statements and then come up with English for complex statements - p= Peyton Manning is great - q = Dolphins will go to the Playoffs Ex: ~q, p ^ q, ~ (p ^ q),… - you can make a truth table to figure out the value of a complex statement based on all possible combinations of the values of simple statement - do a table with p, q, (p V q), (p ^ q), ~(p ^ q), ~(p V q) - do problems like page 5 where you give statements a value and then make sentences with them L.2 Logical Equivalence and Laws of Logic Part 2 all - logically equivalent statements – two statements that have the same truth/false values for all combinations of simple statement values - symbol of equivalence is - ex: p ~(~p) - ex: p ^ q q ^ p - ex: ~(p V q) (~p) ^ (~q) - explain in English - important equivalence statements: - Double Negative (above) - Commutative Property (for ^, V) - DeMorgan’s Law: ~(p V q) ~p ^ ~q and also ~(p ^ q) ( ~p) V (~q) - distributing the “not” flips the conj/disj - Associative Property: p V (q V r) (p V q) V r - same for conjunction - Distributive Property: p ^ (q V r) (p ^ q) V (p ^ r) - and also p V (q ^ r) (p V q) ^ (p V r) - Absorption: p ^ p p, p V p p Ex: do ~(p ^ (~q)) ….. ~p V q [use DeMorgan’s Law - tautology – a statement that is always true for all values of simple statements - ex: p V ~p - contradiction – a statement that is always false for all values of simple statements - ex: p ^ ~p - is the following a contradiction? p ^ ~(p V q)? (yes) - can use Truth tables to prove any tautology - for every combination of simple statements, find both sides of the and make sure they are the same L.3 The Conditional Part 3 all - conditional – statement in the form if ___ then __ - written as p -> Q - pronounced: “p implies Q” - also as “if p, then q” - p is called the hypothesis/premise/given - q is the conclusion - truth table describes how p -> q works (tricky) P Q P -> Q F F T F T T T F F T T T - an implication is only false if the hypothesis is True and conc is False - so if the hypothesis is false, everything is fine and the implication is true (not necessarily the conclusion) - are p->q ~p V q ? - yes since false p is success, other wise q is T…so success - this is an important rule...allows you to switch between implication and disjunctive forms - this is called Equivalent Disjunctive Form (EDF) - given that B is true, what can be said of A -> B? - it has to be true since it can only be false if B is false - biconditional – means “if and only if” (iff) – a double implication where each side implies the other - so A < - > B means A- >B and B->A - write out the Truth table on p8 - has implication, converse, inverse, contrapositive, and biconditional -hw has a lot of truth tables L.4 Five More Laws of Logic - L.4 exs #1, 3-5 (so skip #2) - can use -> to create some new tautologies 1. Simplification: (p ^ q) -> q 2. Law of Syllogism (Transitivity): [(p -> q) ^ (q-> r)] -> (p-> r) - can chain this as much as you like 3. Disjunctive Syllogism: [(p V q) ^ (~p)] -> q 4. Modus Ponens (direct reasoning): [(p->q) ^ p]-> q 5. Modus Tollens (indirect reasoning): [(p->q) ^ ~q] -> ~p - notice how it involved the contrapositive - do examples from p14 where need to determine if it’s MP, MT, or neither - if P is necessary for Q, then Q->P [not assigning these as homework] - if X is sufficient for Y, then X-> Y L.5 Logic Proofs - part 5 all except for #3 (skip it) - premise – statement that we either are given or prove to be true - inference – a maneuver that results in a new premise being made from old ones - argument – list of statements and the conclusion - proof – argument and list of reasons for each step - do 3 problems on p18 - when doing homework, it helps to start out thinking about the conclusion when you are planning strategy L.6 Sets - 2 days part 5 all - see the packet for notes