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Logic Day #2 Math Studies IB NPHS Miss Rose

Logic Day #2 Math Studies IB NPHS Miss Rose. Implications. For two simple propositions p and q , p  q means if p is true, then q is also true. p: it is raining q: I am carrying my polka dot umbrella p  q states: if it is raining then I am carrying my polka dot umbrella.

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Logic Day #2 Math Studies IB NPHS Miss Rose

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  1. Logic Day #2Math Studies IBNPHSMiss Rose

  2. Implications • For two simple propositions p and q, p  q means if pis true, thenq is also true. • p: it is raining • q: I am carrying my polka dot umbrella • p  q states: if it is raining then I am carrying my polka dot umbrella.

  3. Implications are written as and can be read as Implication ‘if ….. then …..’ • If p then q • p implies q • p only if q • p is a sufficient condition for q • q if p • q whenever p

  4. Implications: Truth Table

  5. Consider the following propositions p: It is raining q: I carry an umbrella If it is raining then I carry an umbrella p q T T T T F F F T T F F T p: It is raining q: I carry an umbrella If it is raining then I carry an umbrella p: It is raining q: I carry an umbrella If it is raining then I carry an umbrella p: It is raining q: I carry an umbrella If it is raining then I carry an umbrella T T T T F F The implication is false as it is raining and I am not carrying an umbrella F T T The implication is true, as if it is not raining, I may still be carrying my umbrella. Maybe I think it will rain later, or maybe I am going to use it as a defensive weapon! F F T The implication is true, as if it is not raining, I am not carrying the umbrella

  6. Implications • Yes, this can lead to some “nonsense”-sounding clauses: • If (4 < 3) then (75 > 100) is TRUE • Even some theological quandries • If (1 < 0) then god does not exist is also TRUE • Note, if you make that “if (1 > 0) …” we can’t tell!

  7. Determine whether the statement pq is logically true or false • If 5 * 4 = 20, then the Earth moves around the sun • p is true (5 *4 does = 20) • q is true (The Earth does revolve around the sun) • SO p  q is logically true! • “If NPHS is the Panthers, then Allison is an alien” • p is true (NPHS is the Panthers) • q is false (Allison..?) • SO p  q is logically FALSE!

  8. Determine whether the statement pq is logically true or false • “If Miss Rose has red hair, then Axel is the president” • p is false (Miss Rose doesnot have red hair) • q is false but that doesn’t matter!!!! • SO p  q is logically true!

  9. Converse • The converse is the reverse of a proposition. • The converse of p  q is q  p • p  q states: if it is raining then I am carrying my polka dot umbrella. • q  pstates: if I am carrying my polka dot umbrella then it is raining. • Even if the implication is true, the converse is not necessarily true!!!

  10. INVERSE • If a quadrilateral is a rectangle, then it is a parallelogram • ¬p -> ¬q • If a quadrilateral is not a rectangle then it is not a parallelogram. • Negate both propositions

  11. Contrapositive • If a quadrilateral is a rectangle, then it is a parallelogram • ¬q -> ¬p • If a quadrilateral is not a parallelogram then it is not a rectangle. • Negate both propositions AND change the order • Converse + Inverse = Contrapositive

  12. Equivalent Propositions: • If two combined propositions are true and converse, they are said to be equivalent propositions. • p: Elizabeth is in her math class • q: Elizabeth is F-4 • p  q states: • If Elizabeth is in her math classroom, then she is in F-4 • q  pstates: • If Elizabeth is in F-4, then she is in her math classroom • The two combined statements are both true and converse so they are said to be equivalent • p<-> q

  13. Equivalent Propositions • The truth value of equivalence is true only when all of the propositions have the same truth value.

  14. Consider the following propositions p: I will buy Norma a Mars bar q: She wins the game of Crazy 8’s I will buy Norma a Mars bar if and only if she wins a game of Crazy 8’s p q T T T T F F F T F F F T p: I will buy Norma a Mars bar q: She wins the game of Crazy 8’s I will buy Norma a Mars bar if and only if she wins a game of Crazy 8’s p: I will buy Norma a Mars bar q: She wins the game of Crazy 8’s I will buy Norma a Mars bar if and only if she wins a game of Crazy 8’s p: I will buy Norma a Mars bar q: She wins the game of Crzy 8’s I will buy Norma a Mars bar if and only if she wins a game of Crazy 8’s T T T T F F I brought her the Mars bar even though she didn’t win the game of Crazy 8’s I lied…so the equivalence statement is false. F T F I did not buy Norma the Mar bar so I lied and therefore the equivalence statement is false. F F T The equivalence is true as I did not buy Norma a Mars bar and she did not win Crazy 8’s

  15. Creating longer propositions • When creating truth tables for long propositions, always move from simple  complex • Start with the truth values of each simple proposition • then state any negations • then begin working on compound propositions.

  16. p q T T F F T F T T F T F F F F T F Creating longer propositions • Construct a truth table for

  17. Create a truth table for p p q q T T T T T T T T F F F F F F T T T F F F F F F T p q r T T T T T F T F T F T T T F F F T F F F T F F F T T F F F F T T F F F T T F T F

  18. p p q q T T T T T T T T F F F F F F T T T F F F F F F T Create a truth table for T T F T T F T F F T F T F F T F T T T T T T F T F T F T T F T T T T T F T T F T

  19. p p p q q q T T T T T T T T T p q T T T F F F T F F T T T F F F T T T F T T T F F F F F F F F T F T F T F p F F F T F F T Negation Conjunction Disjunction If you know each of these, you can do any truth table! Equivalence Implication

  20. Translating English Sentences not p it is not the case that p p and q p or q if p then q p implies q if p, q p only if q p is a sufficient condition for q q if p q whenever p q is a necessary condition for p p if and only if q

  21. If there is a thunderstorm then Allison cannot use the computer p: There is a thunderstorm q: Allison uses the computer x is not an even number or a prime number p: x is a even number q: x is a prime number

  22. It is not raining p: It is raining Jesse and Savanna both did the IB Test p: Renzo did the IB Test q: Rafael did the IB Test

  23. If it is raining then I will stay at home. It is raining. Therefore I stayed at home. p: It is raining q: I stay at home If it is raining then I will stay at home If it is raining then I will stay at home. It is raining If it is raining then I will stay at home. It is raining. Therefore I stayed at home.

  24. If I go to bed late then I feel tired. I feel tired. Therefore I went to bed late. p: I go to bed late q: I feel tired If I go to bed late then I feel tired. If I go to bed late then I feel tired. I feel tired. If I go to bed late then I feel tired. I feel tired. Therefore I went to bed late.

  25. I earn money if and only if I go to work. I go to work. Therefore I earn money. p: I earn money q: I go to work I earn money if and only if I go to work. I earn money if and only if I go to work. I go to work. I earn money if and only if I go to work. I go to work. Therefore I earn money.

  26. If I study then I will pass my IB Mathematics. If I pass my IB Mathematics I will get my IB Diploma. I study. Therefore I will get my IB Diploma. p : I Study q : I will pass my IB Mathematics r : I will get my IB Diploma If I study then I will pass my IB Mathematics If I study then I will pass my IB Mathematics. If I pass my IB Mathematics I will get my IB Diploma If I study then I will pass my IB Mathematics. If I pass my IB Mathematics I will get my IB Diploma. I study. If I study then I will pass my IB Mathematics. If I pass my IB Mathematics I will get my IB Diploma. I study. Therefore I will get my IB Diploma.

  27. THE END!!!

  28. Logical Equivalence • There are many different ways to form compound statements from p and q. • Some of the different compound propositions have the same truth values. • In that case, the compound propositions are logically equivalent • EX: ¬p V ¬q and ¬(p^q)

  29. p p p q q q T T T T T T T T T p q T T T F F F F F T T T T F F F T T T T T F T F F F F F F F F T F T F T F p F F F T F F T Negation Conjunction Disjunction Equivalence Implication

  30. Tautology • A tautology is a compound proposition that is always true regardless of the individual truth values of the individual propositions. • A compound proposition is valid if it is a tautology

  31. Contradiction • A contradiction is a compound proposition that is always false regardless of the individual truth values of the individual propositions

  32. Show that the statement is logically valid. • To show that a combined proposition is logically valid, you must demonstrate that it is a tautology. • A tautology is a statement that always tells the truth

  33. p q In order to show that is a tautology we must create a truth table T T T T F F F T T F F T Therefore is logically valid. T F T F F T T T T T T T The statement is true for all truth values given to p and q

  34. p p p q q q T T T T T T T T T p q T T T F F F F F T T T T F F F T T T T T F T F F F F F F F F T F T F T F p F F F T F F T Negation Conjunction Disjunction Equivalence Implication

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