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TRUTH TABLES

TRUTH TABLES. Introduction. Statements have truth values They are either true or false but not both Statements may be simple or compound Compound statements are made up of substatements. Statements. It is raining. The grass is wet. I did my homework. Roses are red. Violets are blue.

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TRUTH TABLES

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  1. TRUTH TABLES

  2. Introduction • Statements have truth values • They are either true or false but not both • Statements may be simple or compound • Compound statements are made up of substatements.

  3. Statements • It is raining. • The grass is wet. • I did my homework. • Roses are red. • Violets are blue.

  4. Compound Statements • Roses are red and violets are blue. • He is very intelligent or he studies at night. • My cat is hungry and he is black.

  5. Questions are not statements • Questions cannot be true or false. • What time is it? • What color is my cat? • What grade will I get in CS230?

  6. TRUTH VALUE • The truth or falsity of a statement is its truth value. • Simple statements have a true or false truth value. • It is raining. T if it is raining F if it isn’t • The truth value of a compound statement is determined by the truth value of the substatements combined with how they are connected.

  7. STATEMENTS • Our book represents statements with the letters • p • q • r • s

  8. COMPOUND STATEMENT • We created compound statements using connectives. • Conjunction (And) • Disjunction (Or) • Negation (Not)

  9. Conjunction • Joining two statements with AND forms a compound statement called a conjunction. • p Λ q Read as “p and q” • The truth value is determined by the possible values of ITS substatements. • To determine the truth value of a compound statement we create a truth table

  10. CONJUNCTION TRUTH TABLE

  11. Conjunction Rule • The compound statement p Λ q will only be TRUE when p is true and q is true

  12. Disjunction • Joining two statements with OR forms a compound statement called a “disjunction. • p ν q Read as “p or q” • The truth value is determined by the possible values of ITS substatements. • To determine the truth value of a compound statement we create a truth table

  13. DISJUNCTION TRUTH TABLE

  14. DISJUNCTION RULE • The compound statement p ν q will only be FALSE when p is false and q is false

  15. NEGATION • ~p read as not p • Negation reverses the truth value of any statement

  16. NEGATION TRUTH TABLE

  17. PROPOSITIONS AND TRUTH TABLES • We can use our connectives to create compound statements that are much more complicated than just 2 substatements. • When p and q become variables of a complex statement we call this a proposition. • ~(pΛ~q) is an example of a proposition • The truth value of a proposition depends upon the truth values of its variables so we create a truth table.

  18. TRUTH TABLE THE PROPOSITION ~(pΛ~q)

  19. PROPOSITIONS AND TRUTH TABLES • First Columns are always your initial variables • 2 variables requires 4 rows • 3 variables requires 8 rows • N variables requires 2n rows • We then create a column for each stage of the proposition and determine the truth value for the stage. • The last column is the final truth value for the entire proposition.

  20. Creating a stepwise truth table

  21. Step 1

  22. Step 2

  23. Step 3

  24. Step 4

  25. TAUTOLOGIES AND CONTRADICTIONS • Tautology – when a proposition’s truth value (last column) consists of only T’s • Contradiction – when a proposition’s truth value (last column) consists of only F’s

  26. Principle of Substitution • If P(p,q,…) is a tautology then P(P1, P2,…) is a tautology for any propositions P1 and P2

  27. Principle of Substitution

  28. LOGICAL EQUIVALENCE • Two propositions P(p,q,…) and Q(p,q, …) are said to be logically equivalent, or simply equivalent or equal when they have identical truth tables. • ~(p Λ q) ≡ ~p V ~q

  29. Logical Equivalence

  30. Conditional and Biconditional Statements • If p then q is a conditional statement • p  q read as p implies q or p only if q • P if and only if q is a biconditional statement • p  q read as p if and only if q

  31. Conditional • p  q

  32. Biconditional • p  q

  33. Conditionals and equivalence~p V q ≡ p  q

  34. Converse, Inverse and Contrapositive

  35. Arguments • An argument is a relationship between a set of propositions P1, P2, … called premises and another proposition Q called the conclusion. • P1, P2, …P8 |- Q • An argument is valid if the premises yields the conclusion • An argument is called a fallacy when it is not valid.

  36. Logical Implication • A proposition P(p,q,…) is said to logically imply a proposition Q(p,q…) written P(p,q…) => Q (p,q…) if Q (p,q…) is true whenever P(p,q…) is true

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