Discrete Mathematics Lecture

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Discrete Mathematics Lecture

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

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