Discrete Mathematics Lecture
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Discrete Mathematics Lecture

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Discrete Mathematics Lecture Presentation Transcript

  • 1. DISCRETE MATHEMATICS
  • 2. LOGIC
  • 3. All Mathematicians wear sandals Anyone who wears sandals is an algebraist Therefore, all mathematicians are algebraist
  • 4. LOGIC
    • Is the study of reasoning
    • Specifically concerned with whether reasoning is correct.
    • Focuses on the relationship among statements as opposed to the content of any particular statement.
  • 5. Propositions
    • Typically expressed as a declarative sentence
    • Basic building blocks of any theory of logic
    • Represented by lowercase letters such as
    • p, q and r.
  • 6. Connectives
    • Used to combine propositions
  • 7. Kinds Of Connectives
    • CONJUCTION – denoted by
    • (read as “p and q”)
    • DISJUNCTION – denoted by
    • ( read as “p or q” )
    • NEGATION -- denoted by
    • (read as “not p”)
  • 8. Kinds Of Connectives
    • CONDITIONAL STATEMENT
          • denoted by p  q
    • (read as If p, then q.)
  • 9. Truth Table Of A Proposition
    • Made up of individual proposition ... , lists all possible combinations of truth values for .... .T denotes true and F denotes false for such combination lists of the truth value of p .
  • 10. CONJUCTION p q T T T T F F F T F F F F
  • 11. DISJUNCTION p q T T T T F T F T T F F F
  • 12. NEGATION p -p q -q T F T F
  • 13. IF-THEN STATEMENTS
      • The most commonly used connectives.
      • It also known as conditional statements or implications.
  • 14. IF-THEN STATEMENTS
      • It consist of the following:
            • Premise – the “if” part
            • Conclusion – the “then” part
      • Represented by the following:
            • If p, then q
            • p -> q
      • Where p and q are the premise and conclusion respectively.
  • 15. IF-THEN STATEMENT
      • Example:
      • If one angle of a triangle is a right triangle , then the other two angles of the triangle are acute angles.
    premise conclusion
  • 16. IF-THEN STATEMENT
      • Example:
      • If one angle of a triangle is a right angle , then the other two angles of the triangle are acute angles.
    p q
  • 17. IF-THEN STATEMENTS
    • it can only be false when the premise is true but the conclusion is false.
  • 18. If
  • 19. If
    • If a picture paints a thousand words
    • Then why can't I paint you?
    • The words will never show
    • For you I've come to know.
    • If a face could launch a thousand ships
    • Then where am I to go?
    • There's no one home but you
    • And now you've left me too.
  • 20.
    • And when my love for life is running dry
    • You come and pour yourself on me
    • If a man could be two places at one time
    • I'd be with you.
    • Tomorrow and today
    • Beside you all the way
    • If the world should stop revolving
    • Spinning slowly down to die.
  • 21.
    • I'd spend the end with you
    • And when the world was through...
    • Then one by one, the stars would all go out.
    • Then you and I, would simply fly away.
  • 22. CONDITIONAL STATEMENT p q p  q T T T T F F F T T F F T
  • 23. BICONDITIONAL STATEMENT
    • It is denoted by :
      • read as “p if and only if q”
  • 24. BICONDITIONAL STATEMENT p q T T T T F F F T F F F T
  • 25. Example:
      • p: Today is Monday.
      • q: it is raining.
      • CONJUNCTION
      • DISJUNCTION
      • NEGATION
      • CONDITIONAL STATEMENT
      • BICONDITIONAL STATEMENT
  • 26. CONJUNCTION
      • p: Today is Monday.
      • q: it is raining.
      • Today is Monday AND it is raining.
  • 27. DISJUNCTION
      • p: Today is Monday.
      • q: it is raining.
      • Today is Monday OR it is raining.
  • 28. NEGATION
      • p: Today is Monday.
      • q: it is raining.
      • -p: Today is NOT Monday.
      • -q: It is NOT raining.
  • 29. CONDITIONAL STATEMENT
      • p: Today is Monday.
      • q: it is raining.
    • p  q
      • IF today is Monday, THEN it is raining.
  • 30. BI CONDITIONAL STATEMENT
      • p: Today is Monday.
      • q: it is raining.
      • Today is Monday IF AND ONLY IF it is raining.