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Logic Math Studies IB NPHS Mrs. Skaff

Logic Math Studies IB NPHS Mrs. Skaff. 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

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Logic Math Studies IB NPHS Mrs. Skaff

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  1. LogicMath Studies IBNPHSMrs. Skaff

  2. 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!

  3. 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: 4 5 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?

  4. 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.

  5. 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.

  6. 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 10ft 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 10ft tall • ¬r: Today is not Sunday

  7. Compound Propositions • Compound propositions are statements which are joined using and or or. • Conjunctions • When two propositions are joined using andthe new proposition is the conjunction of the originals. • The conjunction of the two propositions p and q is denoted by pq

  8. 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.

  9. 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

  10. Conjunction: Truth Value • The truth value of a conjunction is only true with BOTH the propositions are true.

  11. Conjunctions: Truth Value p: Mrs. Skaff is a math teacher q: Mrs. Skaff is 10ft tall r: Today is Sunday • Find the truth value of pq

  12. Simple Truth Table • Find the truth value of p^¬q

  13. Disjunctions • A disjunction is formed when propositions are joined using the word “or” • This is written as pq • The truth value of a disjunction is true when at least one of the propositions is true.

  14. Disjunction • Example: Find the disjunction between the two propositions • p: Pikachu is a Pokemon • q: 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.

  15. 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

  16. 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

  17. 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 False True True False False

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