Introduction to Logic Ch 2 C N Coln

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Introduction to Logic Ch. 2 C. N. Colón St. Barnabas HS Geometry

Introduction to Logic Ch. 2 C. N. Colón St. Barnabas HS Geometry

What is Logic is the science of reasoning. Logic helps us to determine if

What is Logic is the science of reasoning. Logic helps us to determine if a statement is true or false. When this is done we are basically finding its truth value 9/15/2020 2

Definition Deductive Reasoning The process of drawing certain conclusions logically by using an argument.

Definition Deductive Reasoning The process of drawing certain conclusions logically by using an argument. It uses the laws of logic to combine definitions and general statements that we know to be true to reach a valid conclusion. 9/15/2020 3

Definition Inductive Reasoning (found in Ch 3 -1) The method of reasoning in which

Definition Inductive Reasoning (found in Ch 3 -1) The method of reasoning in which a series of particular examples leads to a conclusion 9/15/2020 Free Power. Point Template from www. brainybetty. com 4

A Statement The basic element of logic is the declarative sentence. Remember: sentences that

A Statement The basic element of logic is the declarative sentence. Remember: sentences that are questions, commands, and exclamations, or phrases that are not sentences, are not used in logic. 9/15/2020 5

Examples: Identify each as a true sentence, false sentence, open sentence or not mathematical

Examples: Identify each as a true sentence, false sentence, open sentence or not mathematical at all. Skiing is a summer sport False Softball is a team sport True She is a volleyball player Open Sentence (she) Do you like soccer? Sing that song 2 x – 3 = 5 2 x - 3 9/15/2020 Not a mathematical sentence (question) Not a mathematical sentence (command) Open sentence (variable is x) Not a mathematical sentence (binomial phrase) 6

The negation of a sentence always has the opposite truth value of the given

The negation of a sentence always has the opposite truth value of the given or original statement. Example: A surgeon uses a scalpel It is not true that a surgeon uses a scalpel. A surgeon does not use a scalpel Truth. Table summary: 9/15/2020 p T F True False ~p F T 7

Compound Statements Definition: Statements that are joined by connectives such as “and” (Conjunction) or

Compound Statements Definition: Statements that are joined by connectives such as “and” (Conjunction) or “or” (Dysjunction) Conjunction statements use “and” Example: p: A poodle is a dog q: A bluejay is a bird p q: A poodle is a dog and a bluejay is a bird 9/15/2020 Free Power. Point Template from www. brainybetty. com 8

Conjunction Statements The truth table for a conjunction statement looks like this: p T

Conjunction Statements The truth table for a conjunction statement looks like this: p T T F q T F T p q T F F F If either conjunct is false then the conjunction is false. 9/15/2020 9

Disjunction Statements A disjunction is a compound statement formed by combining two simple statements

Disjunction Statements A disjunction is a compound statement formed by combining two simple statements using the word “or”. Example: p q: 9/15/2020 p: Rena rides the bus to school q: Rena walks to school Rena rides the bus to school or Rena walks to school. 10

Disjunction Statements The truth table for a Disjunction Statement looks like this: p T

Disjunction Statements The truth table for a Disjunction Statement looks like this: p T T F q T F T p q T T T F F F The only case in which the disjunction is false is when both conjuncts or parts are false 9/15/2020 11

Definition Conditional Statement If / then statements are conditional. The then part of the

Definition Conditional Statement If / then statements are conditional. The then part of the statement depends on (is conditional to) the if part. In shorthand, the statement is “if p then q” In symbol form, 9/15/2020 12

There are four statements that can be formed from two simple statements p and

There are four statements that can be formed from two simple statements p and q. The conditional The inverse The contrapositive 9/15/2020 p q ~p ~q q p ~q ~p 13

Definition: Conditional IF something, THEN something else If a car is a Corvette, then

Definition: Conditional IF something, THEN something else If a car is a Corvette, then it is a Chevy If you are studying this topic right now, then you are in F-Period Geometry. 9/15/2020 14

Definition Hypothesis The if part of a conditional statement 9/15/2020 15

Definition Hypothesis The if part of a conditional statement 9/15/2020 15

Definition Conclusion The then part of a conditional statement 9/15/2020 16

Definition Conclusion The then part of a conditional statement 9/15/2020 16

If a car is a Corvette, then it is a Chevrolet Hypothesis (leave out

If a car is a Corvette, then it is a Chevrolet Hypothesis (leave out the “if”) 9/15/2020 Conclusion (leave out the “then”) 17

Susan’s car is a Corvette If a car is a Corvette, then it is

Susan’s car is a Corvette If a car is a Corvette, then it is a Chevrolet Susan’s car is a Chevrolet

The truth table for a conditional statement looks like this: p T T F

The truth table for a conditional statement looks like this: p T T F F 9/15/2020 q T F p q T F T T 19

Definition Inverse: To find the inverse, negate the if and then parts Example: Conditional:

Definition Inverse: To find the inverse, negate the if and then parts Example: Conditional: If a car is a Corvette, then it is a Chevrolet Inverse: If a car is not a Corvette, then it is not a Chevrolet 9/15/2020 20

Definition Converse To write the converse, switch the if and then Example: Conditional: If

Definition Converse To write the converse, switch the if and then Example: Conditional: If a car is a Corvette, then it is a Chevrolet Converse: If a car is a Chevrolet, then it is a Corvette 9/15/2020 Free Power. Point Template from www. brainybetty. com 21

Determine the Converse If you are wearing a skirt, then you are a female

Determine the Converse If you are wearing a skirt, then you are a female If you are a female, then you are wearing a skirt WAIT: What if you are female and not wearing a skirt? We have found a counterexample! 9/15/2020 22

Definition Counterexample A counterexample is an example that proves any statement false. Is there

Definition Counterexample A counterexample is an example that proves any statement false. Is there any female in the room that is not wearing a skirt? Yes, some may be wearing pants. 9/15/2020 23

The Contrapositive Definition: the statement that is formed when you negate the converse Example

The Contrapositive Definition: the statement that is formed when you negate the converse Example Conditional: If Lena combs her hair, then she will look neat p q Converse: q p If she looks neat, then Lena combs her hair Contrapositive: If she does not look neat, then Lena ~q ~p does not comb her hair 9/15/2020 Free Power. Point Template from www. brainybetty. com 24

Biconditional A biconditional statement happens when a conditional statement and its converse are both

Biconditional A biconditional statement happens when a conditional statement and its converse are both true. Silly sounding example: If a figure is a flopper, then it has one eye and two tails If a figure has one eye and two tails, then it is a flopper A figure is a flopper, if and only if it has one eye and two tails

Biconditional Also known as “if and only if” Three ways to represent it: p

Biconditional Also known as “if and only if” Three ways to represent it: p if and only if q p iff q, (that is how it is written) Also written in symbolic form:

Definition If-Then Transitive Property If A then B If B then C You can

Definition If-Then Transitive Property If A then B If B then C You can conclude: If A then C This is also known as a logic chain 9/15/2020 Free Power. Point Template from www. brainybetty. com 27

ASSIGNMENT Complete all of the geometry regents questions on the worksheet that was given

ASSIGNMENT Complete all of the geometry regents questions on the worksheet that was given to you in class today. 9/15/2020 28

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