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Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules AND, OR and NOT Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  1. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Logical Connectives in Daily Life 1. Let’s say we want to fix dinner, a beef dish. 2. Use a search engine to find a recipe. Search term: “beef”. 3. Too many results. Say we want noodles with it. Search term: “beef and noodles”. 4. Still a lot ... maybe with cream sauce. Search term: “beef and noodles and cream”. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  2. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  3. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  4. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  5. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  6. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  7. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  8. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE FALSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  9. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE FALSE FALSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  10. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE FALSE FALSE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  11. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE FALSE FALSE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  12. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE FALSE FALSE TRUE TRUE FALSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  13. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE FALSE FALSE TRUE TRUE FALSE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  14. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE FALSE FALSE TRUE TRUE FALSE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  15. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  16. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  17. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  18. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for AND p q p ∧ q FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE TRUE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  19. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  20. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  21. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  22. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  23. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  24. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  25. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE FALSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  26. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE FALSE FALSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  27. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE FALSE FALSE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  28. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE FALSE FALSE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  29. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE FALSE FALSE TRUE TRUE FALSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  30. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE FALSE FALSE TRUE TRUE FALSE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  31. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE FALSE FALSE TRUE TRUE FALSE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  32. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE FALSE FALSE FALSE TRUE TRUE FALSE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  33. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  34. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  35. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for OR p q p ∨ q FALSE FALSE FALSE FALSE TRUE TRUE TRUE FALSE TRUE TRUE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  36. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for NOT logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  37. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for NOT logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  38. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for NOT p logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  39. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for NOT p ¬ p logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  40. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for NOT p ¬ p FALSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  41. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for NOT p ¬ p FALSE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  42. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for NOT p ¬ p FALSE TRUE TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  43. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules The Truth Table for NOT p ¬ p FALSE TRUE TRUE FALSE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  44. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Visualization With Switches logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  45. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Visualization With Switches 1. Every switch represents a primitive proposition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  46. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Visualization With Switches 1. Every switch represents a primitive proposition. 2. Open switch = FALSE , closed switch = TRUE . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  47. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Visualization With Switches 1. Every switch represents a primitive proposition. 2. Open switch = FALSE , closed switch = TRUE . 3. Compound statement represented by the circuit. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  48. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Visualization With Switches 1. Every switch represents a primitive proposition. 2. Open switch = FALSE , closed switch = TRUE . 3. Compound statement represented by the circuit. 4. Conducting connection from A to B : compound statement TRUE logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  49. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Visualization With Switches 1. Every switch represents a primitive proposition. 2. Open switch = FALSE , closed switch = TRUE . 3. Compound statement represented by the circuit. 4. Conducting connection from A to B : compound statement TRUE , otherwise false. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  50. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Switch Diagram for OR logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  51. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Switch Diagram for OR Disjunction r p � � r ❜ A B r r r q � � r ❜ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  52. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Switch Diagram for AND logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  53. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Switch Diagram for AND Conjunction r r p � q � � � A B r ❜ r ❜ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  54. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Logic Gates logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  55. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Logic Gates 1. Switch diagrams are for explanations only. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  56. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Logic Gates 1. Switch diagrams are for explanations only. 2. But logic circuits are also used in practical applications. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  57. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Logic Gates 1. Switch diagrams are for explanations only. 2. But logic circuits are also used in practical applications. 3. There are three kinds of gates. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  58. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Logic Gates 1. Switch diagrams are for explanations only. 2. But logic circuits are also used in practical applications. 3. There are three kinds of gates. AND-gate logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  59. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Logic Gates 1. Switch diagrams are for explanations only. 2. But logic circuits are also used in practical applications. 3. There are three kinds of gates. AND-gate OR-gate logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  60. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules Logic Gates 1. Switch diagrams are for explanations only. 2. But logic circuits are also used in practical applications. 3. There are three kinds of gates. AND-gate OR-gate NOT-gate (inverter) ❍ ❍ ❍ ✟ r ✟✟ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  61. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  62. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates ... and every computer is essentially a logic circuit. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  63. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates ... and every computer is essentially a logic circuit. How can that be? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  64. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates ... and every computer is essentially a logic circuit. How can that be? Circuit logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  65. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates ... and every computer is essentially a logic circuit. How can that be? Circuit inputs logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  66. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates ... and every computer is essentially a logic circuit. How can that be? Circuit inputs p logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  67. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates ... and every computer is essentially a logic circuit. How can that be? q Circuit inputs p logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  68. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates ... and every computer is essentially a logic circuit. How can that be? r q Circuit inputs p logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  69. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates ... and every computer is essentially a logic circuit. How can that be? r q Circuit outputs inputs p logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  70. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates ... and every computer is essentially a logic circuit. How can that be? r q Circuit outputs inputs p f 1 ( p , q , r ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  71. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates ... and every computer is essentially a logic circuit. How can that be? r q f 2 ( p , q , r ) Circuit outputs inputs p f 1 ( p , q , r ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  72. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates ... and every computer is essentially a logic circuit. How can that be? r f 3 ( p , q , r ) q f 2 ( p , q , r ) Circuit outputs inputs p f 1 ( p , q , r ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  73. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  74. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates If we have two entries u and v , logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  75. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates If we have two entries u and v , 1. ¬ u ∧¬ v is true iff u is false and v is false. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  76. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates If we have two entries u and v , 1. ¬ u ∧¬ v is true iff u is false and v is false. 2. ¬ u ∧ v is true iff u is false and v is true. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  77. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates If we have two entries u and v , 1. ¬ u ∧¬ v is true iff u is false and v is false. 2. ¬ u ∧ v is true iff u is false and v is true. 3. u ∧¬ v is true iff u is true and v is false. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  78. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates If we have two entries u and v , 1. ¬ u ∧¬ v is true iff u is false and v is false. 2. ¬ u ∧ v is true iff u is false and v is true. 3. u ∧¬ v is true iff u is true and v is false. 4. u ∧ v is true iff u is true and v is true. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  79. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates If we have two entries u and v , 1. ¬ u ∧¬ v is true iff u is false and v is false. 2. ¬ u ∧ v is true iff u is false and v is true. 3. u ∧¬ v is true iff u is true and v is false. 4. u ∧ v is true iff u is true and v is true. Similar constructions can be done for more than 2 entries: Put ¬ where you need a FALSE for the output to be true. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

  80. Introduction Truth Tables Switch Diagrams Actual Circuits Arbitrary Connectives Algebraic Rules All Logic Circuits are Constructed from AND-, OR- and NOT-Gates If we have two entries u and v , 1. ¬ u ∧¬ v is true iff u is false and v is false. 2. ¬ u ∧ v is true iff u is false and v is true. 3. u ∧¬ v is true iff u is true and v is false. 4. u ∧ v is true iff u is true and v is true. Similar constructions can be done for more than 2 entries: Put ¬ where you need a FALSE for the output to be true. For every input that gives a true output, construct a statement that is true exactly for that input. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science AND, OR and NOT

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