Geometry

Logic

Logic

What is Logic?

It is our ability to reason.

It is the foundation of sound thinking.

Where do we start?

I start with the question:

What is a sentence?

Every good sentence needs a noun or subject, an action word or a verb and an object or the completion of the idea of the sentence.

How is a mathematical sentence different from an grammatical sentence?

A mathematical sentence has a certain truth value. It is called a closed sentence.

A sentence whose truth value is uncertain is called an open sentence.

Negation

The negation of a sentence changes the truth value of the original sentence.

It is represented by the symbol ~.

If p represents my sentence

then ~p represents it’s negation.

Try This

• In the space provided write a sentence in words.

• Then write the negation of that sentence in words.

Logical Connectors

• A conjunction connects two simple sentences using the word “and “ to make a compound sentence.

• Given the two simple sentences

Sara goes swimming.

Tom is a life guard.

• The conjunction would be

Sara goes swimming and Tom is a life guard

Logical Connectors with symbols

The word And is represented by the symbol

ExampleLet p represent: Sara goes swimming.Let q represent: Tom is a life guard.Sara goes swimming and Tom is a life guard.Would be represented by:

pq

The compound sentence p q is only true when both individual sentences are true.

p q p q

T T T

T F F

F T F

F F F

A disjunction connects two simple sentences using the word “or “ to make a compound sentence.

Given the two simple sentencesSara will studySara goes to the moviesThe compound sentence would be Sara will study or Sara goes to the movies

Disjunction Connective

Using Symbols

The connector “or” is represented by the symbol “.”

Example:Let p represent: Sara will studyLet q represent: Sara goes to the moviesSara will study or Sara will go to the movies.

represents:p q

The disjunction

p q is only false when both p and q are false.

p q p q

T T T

T F T

F T T

F F F

A conditional is a compound sentence usually formed by the words if….then to combine 2 simple sentences. The is used to represent if…then.

• Given the simple sentences

John studies

John gets good grades

• The compound sentence would be:

If John studies then John gets good grades.

Using SymbolsA conditional statement is also called an

implication p implies q.Let p represent: John studiesLet q represent: John gets good grades

If John studies then John gets good grades.Represent by: p q

In a conditional statement the p sentence is called the hypothesis or the premise and the q sentence is called the conclusion.

The conditional statement is only false when a true hypothesis implies a false conclusion.

p q p q

T T T

T F F

F T T

F F T

Homework

In the text bookPg. 41

Numbers 48-61

Pg. 47

Numbers 13-20

Pg. 58

Numbers 3-10