General Education Mathematics

General Education Mathematics

General Education Mathematics

Logic Unit

QUANTIFIED STATEMENTS & NEGATIONS
INSTRUCTIONAL MEDIA / INTERNET VIDEOS
2:25 minutes, Defines simple statements. A statement is a _____ (declarative) sentence that can be classified as true or false.
10:01 minutes, Describes how to form the negations of both universally and existentially quantified statements, Q: One way to form a negation is to put ____ (not) in front of it.
20:30 minutes, First section of logic, negations, qualifiers (all, some, none), negating qualifiers; all are – some are not, none are – some are, some are not – all are, connectives OR, AND, IF/THEN, IF and only IF and their symbols with plenty of examples on a white board, grouping with () on either side of the comma, dominance of connectives, long but thorough.
Q: Sometimes what is not explicitly used in a conditional statement? (then)
PENCASTS
Equivalent and negated quantified statements 3.1.39
PROFESSOR SHOTS LECTURES

Statement, negation, quantified statements, writing equivalent and negations
WRITING ASSIGNMENTS /
  1. What is a statement? Explain why commands, questions, and opinions are not statements.
  2. List the words identified as quantifiers. Give an example of a statement that uses each of these quantifiers.
  3. Evaluate each statement and write exactly what it actually says in simpler language. Then, write the negation of each.
a)All of our fans will not be attending all of our games.
(There will not be any fans at any of the games.)
b)You can’t fool some of the people all of the time.
(There are times when you can’t fool anyone.)
c)I wouldn’t say that everybody doesn’t like my history professor.
(There is at least one person who likes my history professor.)
d)Everyone in this class doesn’t have time to do their homework.
(Nobody in the class has time to do their homework.)
  1. Explain why the negation of “All spring breaks are fun” is not “All spring breaks are not fun.”

COLLABORATIVE & CRITICAL PROBLEM SOLVING /
  1. Use the following pairs of words in quantified statements and draw a diagram of the relationship of each pair. Use at least 4 different types of quantified statements.
a)Humans, mammals
b)Dogs, playful
c)Movies, comedies
d)Mothers, fathers
e)Cubs, World Series winners
f)Poets, writers
COMPOUND STATEMENTS (AND, OR, IF-THEN, IFF)
INSTRUCTIONAL MEDIA / INTERNET VIDEOS
35:42 minutes, Introduction to connectives (and, or, if-then, iff. Writing in symbolic form. Parentheses and dominance of connectives.Long but lots of examples on translation and order of dominance. Q: Look for commas to determine _____ (parentheses).
PENCASTS
Symbolic statement to words, dominance of connectives 3.2.71
PROFESSOR SHOTS LECTURES
Compound statements, and, or, if/then, iff and the alternative ways of stating, dominance of connectives
WRITING ASSIGNMENTS /
  1. Explain the difference between the inclusive and exclusive disjunctions.
  2. Select an advertisement that makes or implies a conditional statement. Analyze the ad to determine whether the consequent necessarily follows from the antecedent. Explain your answer.
  3. Explain the difference between the conditional and bi-conditional statements.
  4. Describe the hierarchy for the basic connectives.

COLLABORATIVE & CRITICAL PROBLEM SOLVING /
  1. Use grouping symbols to clarify the meaning of each symbolic statement.
a)p → q v r ↔ p ʌ r
b)p ʌ q → r ↔ p v r
c)p → p ↔ p ʌ p → ̴p
d)p → p ↔p v p → ̴p
2. Use letters to represent each non-negated simple statement made by well-known people. Then rewrite the given compound statement in symbolic form.
a)“If you cannot get rid of a family skeleton, you may as well make it dance.” (George Bernard Shaw)
b)“I wouldn’t turn out the way I did if I didn’t have all the old-fashioned values to rebel against.” (Madonna)
c)If you know what you believe then it makes it a lot easier to answer questions, and I can’t answer your question.” (George W. Bush)
d)“If you don’t like what you’re doing, you can always pick up your needle and move to another groove.” (Timothy Leary)
TRUTH TABLES (Negation, Conjunction, Disjunction)
INSTRUCTIONAL MEDIA / INTERNET VIDEOS
minutes, An introduction to symbolic logic including defining statements, logical operations and truth tables; review the intro to logic then specifically goes through how to make truth tables. (not the bi-conditional statement though, does not address dominance of connectives)
Q: Negation is a ____(unary) operation which means it operates on a single statement or variable.
6:44 minutes, tautology using truth tables
Q: When the hypothesis is true and the____ (conclusion) is false a conditional statement is false.
PENCASTS
One truth statement using disjunction, negation 3.3.59
PROFESSOR SHOTS LECTURES
Truth tables; negation, conjunction, disjunction and one construction
WRITING ASSIGNMENTS /
  1. Explain the purpose of a truth table.
  2. Describe how to construct a truth table for a compound statement.
  3. What is the difference between a tautology and a self-contradiction?

COLLABORATIVE & CRITICAL PROBLEM SOLVING / The hierarchy of connectives doesn’t distinguish between conjunctions and disjunctions. Does that matter? Construct truth tables for (p v q) ʌ r and p v (q ʌ r) to help you decide.
TRUTH TABLES (Conditional, Bi-Conditional)
INSTRUCTIONAL MEDIA / INTERNET VIDEOS
4:18 minutes, Truth tables for bi-conditional statements
Q: The bi-conditional statement is true only with both ____ (conditional statements) are true.
PENCASTS
Construct a truth table for a compound conditional statement 3.4.79
PROFESSOR SHOTS LECTURES
Construct conditional and bi-conditional truth tables, tautology, implications & self-contradiction
WRITING ASSIGNMENTS / Explain when bi-conditional and conditional statements are true and when they are false.
COLLABORATIVE & CRITICAL PROBLEM SOLVING / Using the (dominance) hierarchy for connectives, write the statement v r by using parentheses to indicate the proper order. Then construct truth tables for () v r and v r). Are the resulting truth values the same? Are you surprised? Why or why not?
CONDITIONAL, CONVERSE, INVERSE, CONTRAPOSITIVE and EQUIVALENT STATEMENTS
INSTRUCTIONAL MEDIA / INTERNET VIDEOS
4:44 minutes, Using truth tables (2 examples) to show that two compound statements are equivalent, Q: How can we show that two statements are equivalent? (Their truth tables match up or are identical)
6:42 minutes, explaining converse, inverse & contrapostive if-then statements, Q: For the ____ (inverse) we negate the hypothesis and conclusion.
PENCASTS
Finding an equivalent statement using a truth table 3.5.15
PROFESSOR SHOTS LECTURES
Converse, inverse, contrapositive and equivalent statements.
WRITING ASSIGNMENTS / 1. Describe how to determine if two statements are equivalent.
2. Describe how to obtain the contrapositive of a conditional statement.
COLLABORATIVE & CRITICAL PROBLEM SOLVING /
  1. Implications restated as Inverse, Converse, and Contrapositive
Each team member must come to class with a logical implication found in a newspaper, magazine, or internet. Reference the location by giving the name and date of publication. Work together to determine the antecedent and the consequent in the implication. State the inverse, converse, and contrapositive of the conditional associated with the implication. Does the contrapositive sound more convincing than the original implication? Why or why not?
  1. Express the statement in “if . . . then” form. Then write the statement’s converse, inverse, and contrapositive. All people who are not fearful are crazy.

NEGATION OF CONDITIONAL STATEMENTS
INSTRUCTIONAL MEDIA / INTERNET VIDEOS
4:29 minutes, The negation of a conditional statement using words (not truth tables) Q: The negation of an If-then statement is not another _____ (if-then)statement.
4:20 minutes, The negation of an if-then statement using a truth table. Q: We can prove the two statements are equal by using a _____ (truth table).
PROFESSOR SHOTS LECTURES
Negations of statements; De Morgan’s Law, quantified all/no, conditional
WRITING ASSIGNMENTS / 1. Explain how to write the negation of a conditional statement.
COLLABORATIVE & CRITICAL PROBLEM SOLVING
DE MORGAN’S LAWS
INSTRUCTIONAL MEDIA / 5:53 minutes, explains de morgan’s laws and shows equivalence using truth tables, Q: We can set up a ______(truth table) to determine the validity of equivalence of two statements.
PENCASTS
Finding an equivalent statement using truth table, contrapositiveDeMorgan’s Law 3.6.39
PROFESSOR SHOTS LECTURES
Use previous lecture
WRITING ASSIGNMENTS /
  1. What are DeMorgan’s Laws and how can they be used?
  2. Explain why the negation of p ʌ q is not ~p ʌ ~q.

COLLABORATIVE & CRITICAL PROBLEM SOLVING / 1. Write a conjunction that is true using the work “all” with both symbols and words. Write the negation of the statement using both forms (words and symbols) using DeMorgan’s Laws.
2. Prove that ~(p v q) ≡ ~p ʌ ~q
FALLACIES and VALID REASONING
INSTRUCTIONAL MEDIA / INTERNET VIDEOS
11:53 minutes, Using truth tables to determine validity, Q: If a statement is not a tautology it is a ____ (fallacy).
Law of syllogism or Law of Transitivity explained well
PENCASTS
Using a truth table to determine validity 3.7.35
PROFESSOR SHOTS LECTURES
Valid arguments, fallacy, standard forms of arguments
WRITING ASSIGNMENTS /
  1. Write an example of the fallacy of the inverse that involves something about your school. Then explain why the conclusion of your arguments is invalid.
  2. Discuss several advertisements and show how logic is used to persuade the reader.
  3. Write an argument matching the law of syllogism (transitive) that involves something about your school. Then explain why the conclusion of your argument is valid.
  4. Write an original argument in words that has a true conclusion, yet is invalid.

COLLABORATIVE & CRITICAL PROBLEM SOLVING /
  1. Politicians argue in favor of positions all the time. Do a Google search for the text of a speech by each of the main candidates in the 2008 Presidential election. Then find at least three logical arguments within the text, write the arguments in symbols, and use truth tables or commonly used argument forms to analyze the arguments, and see if they are valid.
  2. Make up your own example for each of the four common fallacies (Fallacy of the converse, Fallacy of the inverse, Fallacy of the inclusive and, circular reasoning). Use topics that have some relevance to your life.
  3. Make up your own example for each of the four common valid argument forms (Law of detachment, Law of contraposition, Law of syllogism, Law of disjunctive syllogism). Use topics that have some relevance to your life.
  4. Write a valid argument on one of the following questions. If you can, write a valid argument for both sides of the question.
a)Should the death penalty be abolished?
b)Should marijuana be legalized?
c)Should grades be abolished?
d)Should same-sex marriage be legalized?
EULER DIAGRAMS
INSTRUCTIONAL MEDIA / INTERNET VIDEOS
25:28 Minutes, Analyzing Arguments with Euler Diagrams,Q: When you see the word “some” you have ____ (overlapping) circles.
PENCASTS
Using an Euler diagram to analyze arguments 3.8.21
PROFESSOR SHOTS LECTURES
Determining validity of two statements using Euler diagrams
WRITING ASSIGNMENTS /
  1. Explain how to use Euler diagrams to determine whether or not an argument is valid.
  2. Under what circumstances should Euler diagrams, rather than truth tables, be used to determine whether or not an argument is valid?
  3. Explain how to use Euler diagrams to represent the words; some, all, none, some are not.

COLLABORATIVE & CRITICAL PROBLEM SOLVING /
  1. Determine whether the following syllogism is valid or invalid using Euler Diagrams.
All As are Bs.Some Bs are Cs. Therefore,some As are Cs. (Requires 4 diagrams)

Four men, Mr. Baker, Mr. Carpenter, Mr. Draper, and Mr. Smith, live in the same town. One is a baker, one is a carpenter, one is a draper, and one is a smith, but none follows the vocation corresponding to his name. Given that just three of the following four statements are false, find out who is the carpenter. (Mr. Baker is the carpenter)

Mr. Baker is the smith.

Mr. Carpenter is the baker.

Mr. Draper is not the smith.

Mr. Smith is not the draper.

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