Truth tables

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Truth tables

  1. 1. College Math SECTION 3.2: TRUTH TABLES FOR NEGATION, CONJUNCTION, AND DISJUNCTION
  2. 2. Truth Tables <ul><li>A truth table is used to determine when a compound statement is true or false. </li></ul><ul><li>They are used to break a complicated compound statement into simple, easier to understand parts. </li></ul>
  3. 3. Truth Table for Negation <ul><li>As you can see “P” is a true statement then its negation “~P” or “not P” is false. </li></ul><ul><li>If “P” is false, then “~P” is true. </li></ul>P T T F F ~P Case 1 Case 2
  4. 4. Four Possible Cases <ul><li>When a compound statement involves two simple statements P and Q, there are four possible cases for the combined truth values of P and Q. </li></ul>T T T T F F F F P Q Case 1 Case 2 Case 3 Case 4
  5. 5. When is a Conjunction True? <ul><li>Suppose I tell the class, “You can retake the last exam and you can turn in this lab late.” </li></ul><ul><li>Let P be “You can retake the last exam” and Q be “You can turn in this lab late.” </li></ul><ul><li>Which truth values for P and Q make it so that I kept my promise, P Λ Q to the class? </li></ul>
  6. 6. When is a Conjunction True? cont’d. <ul><li>P: “You can retake the last exam.” </li></ul><ul><li>Q: “You can turn this lab in late.” </li></ul><ul><li>There are four possibilities. </li></ul><ul><li>1. P true and Q true , then P Λ Q is true . </li></ul><ul><li>2. P true and Q false , then P Λ Q is false . </li></ul><ul><li>3. P false and Q true , then P Λ Q is false . </li></ul><ul><li>4. P false and Q false , then P Λ Q is false . </li></ul>
  7. 7. Truth Table for Conjunction T T F F T F T F T F F F P Λ Q P Q Case 1 Case 2 Case 3 Case 4
  8. 8. 3.2 Question 1 <ul><li>What is the truth value of the statement, “Caracas is in Venezuela AND Bogota is in Italy”? </li></ul><ul><li>1. True 2. False </li></ul>
  9. 9. When is Disjunction True? <ul><li>Suppose I tell the class that for this unit you will receive full credit if “ You do the homework quiz or you do the lab .” </li></ul><ul><li>Let P be the statement “ You do the homework quiz ,” and let Q be the statement “ You do the lab .” </li></ul><ul><li>In this case a “truth” is equal to receiving full credit </li></ul>
  10. 10. When is Disjunction True? cont’d. <ul><li>P: “You do the homework quiz.” </li></ul><ul><li>Q: “You do the lab.” </li></ul><ul><li>There are four possibilities: </li></ul><ul><li>1. P true and Q true , then P V Q is true . </li></ul><ul><li>2. P true and Q false , then P V Q is true . </li></ul><ul><li>3. P false and Q true , then P V Q is true . </li></ul><ul><li>4. P false and Q false , then P V Q is false . </li></ul>
  11. 11. Truth Table for Disjunction T T F F T F T F T T T F P V Q P Q Case 1 Case 2 Case 3 Case 4
  12. 12. 3.2 Question 2 <ul><li>What is the truth value of the statement, “Caracas is in Venezuela or Bogota is in Italy”? </li></ul><ul><li>1. True 2. False </li></ul>
  13. 13. Truth Table Summary <ul><li>You can remember the truth tables for ~ (not) , </li></ul><ul><li>Λ (and) , and, V (or) by remembering the following: </li></ul><ul><li>~(not) - Truth value is always the opposite </li></ul><ul><li>Λ (and) - Always false, except when both are true </li></ul><ul><li>V (or) - Always true, except when both are false </li></ul>
  14. 14. Making a Truth Table Example <ul><li>Let’s look at making truth tables for a statement involving only ONE Λ or V of simple statements P and Q and possibly negated simple statements ~P and ~Q. </li></ul><ul><li>For example, let’s make a truth table for the statement ~P V Q </li></ul>
  15. 15. Truth Table for ~P V Q T T F F T F T F P ~P Q Q Opposite of Column 1 F F T T Same as Column 2 T F T F T F T T Final Answer column V
  16. 16. Another Example: P Λ ~Q T T F F T F T F P P Q ~Q Same as Column 1 T T F F Opposite of Column 2 F T F T F T F F Final Answer column Λ
  17. 17. 3.2 Question 3 <ul><li>What is the answer column in the truth table of the statement ~P Λ ~Q ? </li></ul><ul><li>1. T 2. T 3. F </li></ul><ul><li> F F F </li></ul><ul><li> F T F </li></ul><ul><li>F F T </li></ul>
  18. 18. ~P Λ ~Q Stop Day 1 T T F F T F T F P ~P Q ~Q Opposite of Column 1 F F T T Opposite of Column 2 F T F T F F F T Final Answer column Λ
  19. 19. More Complicated Truth Tables <ul><li>Now suppose we want to make a truth table for a more complicated statement, </li></ul><ul><li>(P V ~Q) V (~P Λ Q) </li></ul><ul><li>We set the truth table up as before. </li></ul><ul><li>Our final answer will go under the most dominant connective not in parentheses </li></ul><ul><li>( the one in the middle ) </li></ul>
  20. 20. More Complicated Truth Tables Final Answer T T F F Opposite of Column 1 Opposite of Column 2 Same as Column 2 Same as Column 1 F T F T OR T T F T F F T T T F T F AND F F T F T T T T
  21. 21. More Complicated Truth Tables <ul><li>Now let’s make a truth table for </li></ul><ul><li>(P V ~Q) Λ (~P Λ Q) </li></ul><ul><li>Each of the statements in parentheses </li></ul><ul><li>( P V ~Q) and (~P Λ Q) are just like the statements we did previously, so we fill in their truth tables as we just did. </li></ul>
  22. 22. More Complicated Truth Tables Final Answer T T F F Opposite of Column 1 Opposite of Column 2 Same as Column 2 Same as Column 1 F T F T OR T T F T F F T T T F T F AND F F T F F F F F P Q ( P ~Q ) ( ~P Q ) T T T F F T F F
  23. 23. Constructing Truth Tables with Three Simple Statements <ul><li>So far all the compound statements we have considered have contained only two simple statements (P and Q), with only four true-false possibilities. </li></ul>P Q Case 1 T T Case 2 T F Case 3 F T Case 4 F F
  24. 24. Constructing Truth Tables with Three Simple Statements cont’d. <ul><li>When a compound statement consists of three simple statements (P, Q, and R), there are now eight possible true-false combinations. </li></ul>
  25. 25. Constructing Truth Tables with Three Simple Statements cont’d. P Q R Case 1 T T T Case 2 T T F Case 3 T F T Case 4 T F F Case 5 F T T Case 6 F T F Case 7 F F T Case 8 F F F
  26. 26. A Three Statement Example <ul><li>Lets construct a truth table for the statement (P V Q) Λ ~R using the same techniques as before. </li></ul><ul><li>Remember, there are not more possible combinations because we added a third statement </li></ul>
  27. 27. A Three Statement Example T T T T F F F F T T F F T T F F F T F T F T F T T T T T T T F F F T F T F T F F P Q R (P Q) ~R T T T T T F T F T T F F F T T F T F F F T F F F Final Answer
  28. 28. Practice <ul><li>Determine the Truth Value for the statement IF: </li></ul><ul><ul><ul><li>P is true, Q is false, and R is true </li></ul></ul></ul><ul><li>(~ P V ~ Q) Λ ( ~R V ~ P) </li></ul>
  29. 29. Practice <ul><li>Translate into symbols. Then construct a truth table and indicate under what conditions the compound statement is TRUE. </li></ul><ul><li>Tanisha owns a convertible and Joan does not own a Volvo. </li></ul>
  30. 30. Practice <ul><li>Construct a Truth Table for the following compound statement: R V(P Λ ~ Q) </li></ul>
  31. 31. DeMorgans Law (this guy again?)
  32. 32. More Complicated Truths; Quantifiers <ul><li>Quantifiers- Give an Amount to a statement </li></ul><ul><li>Examples; </li></ul><ul><ul><ul><li>All </li></ul></ul></ul><ul><ul><ul><li>No/None </li></ul></ul></ul><ul><ul><ul><li>Some </li></ul></ul></ul><ul><ul><ul><li>Half </li></ul></ul></ul><ul><ul><ul><li>At least one </li></ul></ul></ul><ul><li>This makes a Negation (~) more difficult to define </li></ul><ul><ul><li>Find the Negation of; </li></ul></ul><ul><ul><ul><li>Some Do </li></ul></ul></ul><ul><ul><ul><li>All do </li></ul></ul></ul><ul><ul><ul><li>None do </li></ul></ul></ul><ul><ul><ul><li>At least one </li></ul></ul></ul>
  33. 33. Negations of Quantifiers <ul><li>Some do </li></ul><ul><li>All do </li></ul><ul><li>None do </li></ul><ul><li>At least one does </li></ul><ul><li>None do (All do not) </li></ul><ul><li>Some do Not (Not all do) </li></ul><ul><li>Some do (None do not) </li></ul><ul><li>None do </li></ul>
  34. 34. Examples of Negations with quantifiers <ul><li>Some girls play soccer </li></ul><ul><li>All boys are immature </li></ul><ul><li>No students read books </li></ul><ul><li>At least one person likes anchovies </li></ul><ul><li>No Girls play soccer </li></ul><ul><li>Not all boys are immature (some are not immature) </li></ul><ul><li>Some students read books </li></ul><ul><li>No one likes anchovies </li></ul>

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