LogicMath Studies IB

NPHSMrs. Skaff

Symbol

In Words Meaning

p, q, or r

¬

What is Logic?Logic is a way to describe situations or

knowledge that enables us to reason from existing knowledge to new conclusions.

For example:All college students are poorI am college a studentTherefore, I am poor

If the original statement is false, the conclusion is still logical (it just may be false)All college students are rich (not true)I am a college studentTherefore, I am rich!

Logical ReasoningYou have four cards. A letter A – Z is the blue

side. A number 0 – 9 is on the green side. You have these cards:

E K 4 7

What card do you turn over to test the rule “If a card has a vowel on the blue side, it must have an even number on the greenside?

5

Sets and Logical ReasoningA proposition is a statement that may be

true or false.For Example,

“Mrs. Skaff is a math teacher” is a true proposition

“Mrs. Skaff is 10 feet tall” is a false proposition“Today is Saturday” is indeterminate because

it may be true of false depending on the circumstances.

All of these are examples of simple propositions.

PropositionsConsider the following statements:Go get the bookHave you seen my new shirt?The dog is behind the shed

Which statement is a proposition?The dog is behind the shed.

Rewrite the others so they are propositionsSiya got the book.Jackson saw my new shirt.

Representing propositionsWe represent propositions by letters such as

p, q, and r.For Example:

p: Mrs. Skaff is a math teacherq: Mrs. Skaff is 10ft tallr: Today is Sunday

Opposites are represented by negation (¬)¬p represents the opposite of pWe read that as “not p”

¬p: Mrs. Skaff is not a math teacher ¬q: Mrs. Skaff is not 10ft tall ¬r: Today is not Sunday

Compound PropositionsCompound propositions are statements

which are joined using and or or. Conjunctions

When two propositions are joined using and the new proposition is the conjunction of the originals.

The conjunction of the two propositions p and q is denoted by pq

Truth ValuesBecause a proposition is a statement that can

be either true or false, its Truth Values are T for true and F for false. This may be represented in a table.p : Carla has

black hair¬ p : Carla does not have black hair

T

F

ConjunctionsFor the following propositions:

p: Ty is a superheroq: Sofia is a basketball star

What is p^q : Ty is a superhero and Sofia is a

basketball star What is ¬p^q

Ty is not a superhero and Sofia is a basketball star

Conjunction: Truth ValueThe truth value of a conjunction is only true

with BOTH the propositions are true.

p q pq

T T

T F

F T

F F

Conjunctions: Truth Valuep: Mrs. Skaff is a math teacherq: Mrs. Skaff is 10ft tallr: Today is SundayFind the truth value of pq

p q pq

T T T

T F F

F T F

F F F

Simple Truth Tablep q ¬q p¬q

T T F F

T F T T

F T F F

F F T F

• Find the truth value of p^¬q

DisjunctionsA disjunction is formed when propositions

are joined using the word “or” This is written as p q

The truth value of a disjunction is true when at least one of the propositions is true.

p q p q

T T T

T F T

F T T

F F F

DisjunctionExample: Find the disjunction between the

two propositionsp: Pikachu is a Pokemonq: Mrs. Dogancay is a Pokemon

p is true (Mrs. Skaff’s favorite pokemon!)q is not true (unless there is something Mrs.

Dogancay is hiding from us)THEREFORE, p q is true.p q p ∨ q

T T T

T F T

F T T

F F F

Exclusive DisjunctionChandler will go home if John is running late or

if it is raining. You will go to Hawaii by boat or by plane.In the first, Chandler will go home if John is

late, it’s raning, or bothIn the second, you can travel by boat or plane,

but not both at the same time.Thi is called an exclusive disjunction

p q p q p q

T T F

T F T

F T T

F F F

Venn DiagramsYou will go to Hawaii by boat or by plane

p: you will go by boatq: you will go by plane

planeboat

Practicep: London is the capital of England

q: 45 – 5 = 4r: Cows have 4 legs

Find the truth value of the following:p ^ qq v r¬q ^ r¬(p v q)p v r

FalseTrueTrueFalse

False