Truth Tables Lecture
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Questions and Answers

What is the main purpose of a truth table?

  • To prove an argument is valid
  • To test the validity of an argument
  • To deduce the conclusion of an argument
  • To visualize the logical relationships between statements (correct)
  • What can be inferred from the truth table for ~p ∧ q?

  • p must be false and q must be true (correct)
  • q must be true and p must be false
  • p and q can never both be true
  • p and q must always be true
  • What is the logical equivalent of ~(~p)?

  • ~p
  • p ∧ q
  • p (correct)
  • p ∨ q
  • What can be concluded about ~(p ∧ q) and ~p ∧ ~q?

    <p>They are never logically equivalent</p> Signup and view all the answers

    What is the purpose of De Morgan's Laws?

    <p>To convert between conjunctive and disjunctive normal forms</p> Signup and view all the answers

    What can be inferred from the truth table for (p ∨ q) ∧ ~(p ∧ q)?

    <p>The statement is true when p and q have different truth values</p> Signup and view all the answers

    What is the negation of an 'and' statement equivalent to?

    <p>An 'or' statement with negated components</p> Signup and view all the answers

    What can be concluded about the statement 'It is not true that I am not happy'?

    <p>It is logically equivalent to 'I am happy'</p> Signup and view all the answers

    What is the symbolic representation of the negation of an 'and' statement?

    <p>~(p ∧ q) ≡ ~p ∨ ~q</p> Signup and view all the answers

    What is the negation of the statement 'Akram is unfit and Saleem is injured'?

    <p>Akram is not unfit or Saleem is not injured</p> Signup and view all the answers

    What is the negation of the statement '-1 < x ≤ 4'?

    <p>x ≤ –1 or x &gt; 4</p> Signup and view all the answers

    What is the definition of a tautology?

    <p>A statement form that is always true regardless of the truth values of the statement variables</p> Signup and view all the answers

    What is the negation of the statement 'p ∧ (q ∧ r)'?

    <p>~(p ∧ (q ∧ r)) ≡ ~p ∨ ~q ∨ ~r</p> Signup and view all the answers

    What is the symbolic representation of the negation of an 'or' statement?

    <p>~(p ∨ q) ≡ ~p ∧ ~q</p> Signup and view all the answers

    What is the negation of the statement 'The fan is slow or it is very hot'?

    <p>The fan is not slow and it is not very hot</p> Signup and view all the answers

    Are the statements '(p ∧ q) ∧ r' and 'p ∧ (q ∧ r)' logically equivalent?

    <p>Yes</p> Signup and view all the answers

    Study Notes

    Truth Tables

    • Truth table for ~p ∧ q:
      • ~p ∧ q is false when p is true
      • ~p ∧ q is true when p is false and q is true
    • Truth table for ~p ∧ (q ∨ ~r):
      • ~p ∧ (q ∨ ~r) is true when p is false and q is true or r is false
      • ~p ∧ (q ∨ ~r) is false when p is true or q is false and r is true
    • Truth table for (p ∨ q) ∧ ~(p ∧ q):
      • (p ∨ q) ∧ ~(p ∧ q) is true when p is true and q is false, or p is false and q is true
      • (p ∨ q) ∧ ~(p ∧ q) is false when p and q are both true or both false

    Double Negative Property

    • ~(~p) ≡ p (Double Negative Property)
    • Example: "It is not true that I am not happy" is equivalent to "I am happy"

    DeMorgan's Laws

    • ~(p ∧ q) ≡ ~p ∨ ~q (DeMorgan's Law)
    • ~(p ∨ q) ≡ ~p ∧ ~q (DeMorgan's Law)
    • Example: ~(p ∧ q) is not logically equivalent to ~p ∧ ~q
    • Application: Give negations for each of the following statements:
      • a. The fan is not slow and it is not very hot
      • b. Akram is not unfit or Saleem is not injured

    Inequalities and DeMorgan's Laws

    • Use DeMorgan's Laws to write the negation of -1 < x ≤ 4
    • The negation is: x ≤ –1 or x > 4

    Tautology

    • A tautology is a statement form that is always true regardless of the truth values of the statement variables
    • Example: p ∨ ~p is a tautology
    • Truth table for p ∨ ~p:
      • p ∨ ~p is always true

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    Description

    This quiz covers truth tables for various logical statements, including ~p∧q, ~p ∧ (q ∨ ~r), and (p∨q) ∧ ~(p∧q).

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