ch. 2-logic day 1

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22-- Reasoning & ProofReasoning & ProofTodays topic: LogicTodays topic: LogicCAUTION: This unit requires you to open your minds

and think critically and creatively.You will NOT always have step-by-step instructions.

Problems WILL have more than one solution.

Complete attention in class is necessary to understand

these abstract concepts.

I will do my best to help you, but you will have to do

your part and complete all homework and participate in

class.

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A few important definitionsA few important definitions

y Inductive Reasoning is reasoning based

on patterns you observe.

y A conjecture is a conclusion you reach

using inductive reasoning.

y A counterexample is an example for

which the conjecture is incorrect.

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How does logic apply to school?How does logic apply to school?

y When you answer true-false questions on a test, you areusing a basic principle of logic. For example, refer to themap, and answer true or false.

y Raleigh is a city in North Carolina.

y You know that there is only one correct answer, either trueor false.

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Determine Truth ValuesDetermine Truth Values

y A statement, like the true-false

example before, is any sentence that is

either true or false, but not both.

Unlike a conjecture, we know that a

statement is either true or false.

The truth or falsity of a statement is called its

truth value.

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Determine Truth ValuesDetermine Truth Values

y Statements are often represented using aletter such as p or q. p: Raleigh is a city in North Carolina.x This statement is true.

y The negation of a statement has theopposite meaning as well as an oppositetruth value. For example, the negation ofthe statement above is not p (in symbols, ~p).

not p (~p): Raleigh is not a city in North Carolina.x In this case, the statement is false.

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Compound StatementsCompound Statements

y Two or more statements can be joined to forma compound statement. Consider thefollowing two statements.

p: Raleigh is a city in North Carolina. q: Raleigh is the capital of North Carolina.

y The two statements can be joined by the wordand(in symbols, pq).

p andq (pq ): Raleigh is a city in North Carolina,andRaleigh is the capital of North Carolina.

y A conjunction is true only when both statements init are true.

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Truth Values of ConjunctionsTruth Values of ConjunctionsUse the following statements to write a compound statement for each

conjunction. Then find its truth value.

y p: January 1 is the first day of the year.

y q: -5 + 11 = -6

y r: A triangle has three sides

y pq January is the first day of the year, and -5 + 11 = -6

pq is false, because p is true and q is false.

y rp A triangle has three sides, and January 1 is the first day of the year.

rp is true, because r is true and p is true.

y p~r January 1 is the first day of the year, and a triangle does not have three sides.

p~r is false, because p is true and ~r is false.

y ~qr -5 + 11 { -6, and a triangle has three sides.

~q

r is true because ~q is true and r is true.

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DisjunctionDisjunction

y Statements can also be joined by the word or.This type of statement is a disjunction.Consider the following statements. p: Ahmed studies chemistry.

q: Ahmed studies literature.

p or q (pq): Ahmed studies chemistry orAhmedstudies literature.

y A disjunction is true if at least one of the

statements is true. In the case of p or q above,the disjunction is true if Ahmed either studieschemistry or literature or both. The disjunctionis false only if Ahmed studies neither chemistrynor literature.

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Truth Values ofDisjunctionsTruth Values ofDisjunctionsUse the following statements to write a compound statement for

each disjunction. Then find its truth value.

y p: 100 z 5 = 20

y q: The length of a radius of a circle is twice the length of itsdiameter.

y r: The sum of the measures of the legs of a right triangle equals themeasure of the hypotenuse.

y pq

100 z 5 = 20, or the length of a radius of a circle is twice the length ofits diameter.

pq is true, because p is true. It does not matter that q is false.

y q r

The length of a radius of a circle is twice the length of its diameter, orthe sum of the measures of the legs of a right triangle equals themeasure of the hypotenuse.

q r is false since neither statement is true.

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Truth TablesTruth Tables

y A convenient method for organizing the

truth values of statements is to use a

truth table.

p q ~p ~q pq qp

T T F F T T

T F F T F TF T T F F T

F F T T F F

Follow the pattern in the first

two columns for all truth tables.

A conjuction is true only when

both statements are true.

A disjunction is false only when

both statements are false.

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Construct Truth TablesConstruct Truth Tables

y Construct a truth table for the compound

statement.

y p~q

p q ~q p~q

T T F F

T F T T

F T F F

F F T F

Step 1: Make columns with the

headings p, q, ~q, and p

~qStep 2: List the possible

combinations of truth values for p and

q using the pattern shown here.

Step 3: Use the truth values of q to

determine the truth values of~q.Step 4: Use the truth values for p

and ~q to write the truth values for

p~q

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y ~p~q

p q ~p ~q ~p~q

T T F F F

T F F T T

F T T F T

F F T T T

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y (pq) r

p q r pq (pq) r

T T T T T

T T F F T

T F T T T

T F F F F

F T T F T

F T F F T

F F T F F

F F F F F

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HomeworkHomework

y 2-2 Logic Practice Worksheet that I will pass out. All problems on the page

Reminder: these problems are testing your logic skills,use your logic skills that you walked in with.

Attempt ALL problems, whether you understand them ornot. You definitely will not understand if you dont writeanything down.

If you are still confused, come see me before or afterschool, but dont get behind and then expect to catch up

the day before the test.y This powerpoint can be found on my website for

you to use to help you complete your homework.

ch. 2-logic day 1

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