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CHAP 1: Logic

CHAP 1: Logic. “If it was so, it might be: and if it were so, it would be; but as it isn’t, it ain’t. That’s logic” Lewis Carroll, author of Alice in Wonderland. Part 1: Statements & Standard Forms. Statement:

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CHAP 1: Logic

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  1. CHAP 1: Logic “If it was so, it might be: and if it were so, it would be; but as it isn’t, it ain’t. That’s logic” Lewis Carroll, author of Alice in Wonderland

  2. Part 1: Statements & Standard Forms • Statement: • Any declarative sentence that is either true (T) or false (F) but not both • T or F are called truth value of the statement • Statements use lower case letters like p, q, r

  3. EXAMPLES OF STATEMENTS • Let p be the sentence “Santa is Jolly” • Let q be the sentence “ Christmas is on July 4th” • Number 1 & 2 are statements bc we can assign to them a truth value • Is the following a statement? • “Solve the following for X” • THE ABOVE is a COMMAND

  4. More Words • Negation • The statement whose truth value is the opposite of the truth value of p • The symbol for a negation is ~ • When you have ~ p it is read as “not p”

  5. EXAMPLES OF NEGATIONSTATEMENTS • Let p be the sentence “Santa is Jolly” • What would be a ~p ? • “Santa is not jolly” • Let q be the sentence “ Christmas is on July 4th” • What would be a ~q ? • “Christmas is not on July 4th” • Notice that these negations are also statements that can be given truth values

  6. Truth Tables • Used to show the truth values for different statements • Each Column represents a statement • And the rows below show the different truth values possible

  7. QUICK NOTE • The negations of statements will be referred to as being in STANDARD FORM • Example: • the standard form of the negation of the statement “Santa is jolly” is the statement “Santa is not jolly” • The standard form of the negation of the statement “ Texas is not a state” is ……. • “Texas is a state”

  8. YOUR TURN to TRY • Write a negation in standard for each of the statements • a: Dallas is the capital of Oklahoma • b: An orange is a fruit • c: An apple is not Blue

  9. Check YoUrAnSwErS • ~a: Dallas is not the capital of Oklahoma • ~b: An orange is not a fruit • ~c: An apple is Blue

  10. Now Lets Give the Each Statement a Truth Value a __________ ~ a _________ a: Dallas is the capital of Oklahoma b: An orange is a fruit c: An apple is not Blue b __________ ~ b __________ ~a: Dallas is not the capital of Oklahoma ~b: An orange is not a fruit ~c: An apple is Blue c __________ ~ c _________

  11. Another New Word • Conjunction • Is putting two statements together using the word and • The symbol for and is ^ • Example p ^ q is read as “p and q” • This statement is true ONLY if BOTH are true

  12. EXAMPLES OF CONJUNCTION STATEMENTS • Let p be the sentence “Santa is Jolly” • Let q be the sentence “ Christmas is on July 4th” • Let r be the sentence “George Washington was the first U.S. President: • Write a conjunction of p and q • Write a conjunction of p and r • Write a conjunction of q and r

  13. Check your answers • Santa is jolly and Christmas is on July 4th • Santa is jolly and George Washington was the first U.S. President • Christmas is on July 4th and George Washington was the first U.S. President

  14. What are the Truth Values? • Santa is jolly and Christmas is on July 4th • FALSE! FALSE! • Santa is jolly and George Washington was the first U.S. President • TRUE! TRUE! TRUE! • Christmas is on July 4th and George Washington was the first U.S. President • FALSE!!! FALSE!

  15. A General Truth Table for p ^ q

  16. One Last Word • DISJUNCTION • Is putting two statements using the word or • The symbol for orisv • Example p v q is read as “ p or q” • The statement is true IF and ONLY IF at LEAST ONE of p and q is TRUE

  17. EXAMPLES OF DISJUNCTION STATEMENTS • Let p be the sentence “Santa is Jolly” • Let q be the sentence “ Christmas is on July 4th” • Let r be the sentence “George Washington was the first U.S. President: • Write a disjunction of p and q • Write a disjunction of p and r • Write a disjunction of q and r

  18. Check your answers • Santa is jolly or Christmas is on July 4th • Santa is jolly or George Washington was the first U.S. President • Christmas is on July 4th or George Washington was the first U.S. President

  19. What are the Truth Values? • Santa is jolly or Christmas is on July 4th • TRUE! • Santa is jolly or George Washington was the first U.S. President • TRUE! TRUE! TRUE! • Christmas is on July 4th and George Washington was the first U.S. President • TRUE! TRUE! TRUE!

  20. WHAT ABOUT………..????? • ~ p v q

  21. A General Truth Table for p v q

  22. NOW time to PUT IT all Together • Given Statements • a: Abraham Lincoln was the 16th U.S. president • b: A lemon is tart • c: Valentines Day is in December

  23. Questions • a ^ b • a v c • ~a ^ b • ~ c v ~ a • The truth value for # 1 is ____________ • The truth value for # 2 is ____________ • The truth value for # 3 is ____________ • The truth value for # 4 is ____________

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