Logic Math Studies IB NPHS Mrs Skaff Symbol
Logic Math Studies IB NPHS Mrs. Skaff
Symbol p, q, or r ¬ In Words Meaning
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 college a 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) I am a college student 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 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?
Sets and Logical Reasoning A 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.
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 Siya 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. Skaff is a math teacher q: Mrs. Skaff is 10 ft tall r: Today is Sunday Opposites are represented by negation (¬) ¬p represents the opposite of p We read that as “not p” ¬p: Mrs. Skaff is not a math teacher ¬q: Mrs. Skaff is not 10 ft tall ¬r: Today is not Sunday
Compound Propositions
Truth Values 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 T F ¬ p : Carla does not have black hair
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 T T F q T F F
Conjunctions: Truth Value p q T T F F F T F F
Simple Truth Table p q ¬q T T F F F F T F • Find the truth value of p^¬q
Disjunctions p q T T F F F
Disjunction p q p∨q T T F F F
Exclusive Disjunction p q T T F T F T T F F 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 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
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