Logic Math Studies IB Propositions, truth tables, logic symbols, negation, conjunction, disjunction.

Formula Booklet page 4

Symbol In Words Meaning p, q, or r propositions Statements with truth value assigned ¬ negation NOT ∧ conjunction AND ∨ Inclusive disjunction OR Exclusive disjunction OR but not both implication IF … THEN

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 poor I am a college student Therefore, 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) Therefore, I am rich!

Logical Reasoning You have four cards. A letter A – Z is the blue side. A number 0 – 9 is on the green side. You have these cards: 5 4 7 E K 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?

Propositions.

Sets and Logical Reasoning A proposition is a statement that may be true or false. For Example, “Mrs Graham is a maths teacher” is a true proposition “Mrs Graham is 2 meters 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.

Propositions Consider the following statements: Go get the book Have 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 propositions Sophia got the book. Jackson saw my new shirt.

Representing propositions We represent propositions by letters such as p, q, and r. For Example: p: Mrs Graham is a math teacher q: Mrs Graham is 2 m tall r: Today is Sunday Opposites are represented by negation (¬) ¬p represents the opposite of p We read that as “not p” ¬p: Mrs Graham is not a math teacher ¬q: Mrs Graham is not 2 m tall ¬r: Today is not Sunday

A proposition is a statement that is either true (T) or false (F); it cannot be both true and false at the same time. Which of the following are propositions? Riley studies IB mathematics. 5 < 6 Have a nice day! Is it snowing? The sun is not shining. The moon is made of green cheese. A week consists of five days. The glass is full. The air-con is on.

Write the negation of the statement ‘n is a prime number’ p: n is a prime number ¬ p: n is not a prime number Write the negation of ‘this word starts with a vowel’ without using the word ‘not’. ‘this word starts with a consonant’

Truth Values p : Carla has black hair Because 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

Truth table for negation

Exercise 8A.1 p 233 Questions 1-5 Exercise 8A.2 Questions 1-4

Compound Propositions Compound 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 p∧q

Compound propositions. A compound proposition is made up of simple propositions joined together by connectives, The five connectives that we will use most commonly are: NOT AND OR IF … THEN

Conjunctions For the following propositions: p: Ty is a superhero q: 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 Value The truth value of a conjunction is only true with BOTH the propositions are true. p q p∧q T F

Conjunctions: Truth Value p: Mrs Graham is a math teacher q: Mrs Graham is 2 m tall r: Today is Sunday Find the truth value of p∧q p q p∧q T F

Simple Truth Table p q ¬q p∧¬q T F Find the truth value of p^¬q

Complete the truth table for is a logical contradiction because all the entries in the third column are false.

Disjunctions A 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 F

Disjunction Example: Find the disjunction between the two propositions p: Pikachu is a Pokemon q: Mrs. Dogancay is a Pokemon p is true q is not true (unless there is something Mrs. Dogancay is hiding from us) THEREFORE, p ∨ q is true. p q p ∨ q T F

Exclusive Disjunction Chandler 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 both In 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 F

Venn Diagrams You will go to Hawaii by boat or by plane p: you will go by boat q: you will go by plane boat plane

Practice False True True False False p: London is the capital of England q: 45 – 5 = 4 r: Cows have 4 legs Find the truth value of the following: p ^ q q v r ¬q ^ r ¬(p v q) p v r False True True False False