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Elementary Mathematics

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MTH001 ­ Elementary Mathematics
LECTURE # 5
EXAMPLE
An interesting teacher keeps me awake. I stay awake in Discrete Mathematics class.
Therefore, my Discrete Mathematics teacher is interesting.
Is the above argument valid?
ARGUMENT:
An argument is a list of statements called premises (or assumptions or
hypotheses) followed by a statement called the conclusion.
P1  Premise
P2 Premise
P3  Premise
. . . . .. . . . .
Pn  Premise
______________
C  Conclusion
NOTE The symbol \ read "therefore," is normally placed just before the conclusion.
VALID AND INVALID ARGUMENT:
An argument is valid if the conclusion is true when all the premises are true.
Alternatively, an argument is valid if conjunction of its premises imply conclusion.That is
(P1P2 P3 . . . Pn) C is a tautology.
An argument is invalid if the conclusion is false when all the premises are true.
Alternatively, an argument is invalid if conjunction of its premises does not imply conclusion.
EXAMPLE:
Show that the following argument form is valid:
pq
p
 q
SOLUTION
premises
conclusion
pq
p
q
p
q
critical row
T
T
T
T
T
T
F
F
T
F
F
T
T
F
T
F
F
T
F
F
EXAMPLE
Show that the following argument form is invalid:
pq
q
 p
Page 19
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MTH001 ­ Elementary Mathematics
SOLUTION
premises
conclusion
pq
p
q
q
p
T
T
T
T
T
T
F
F
F
T
critical row
F
T
T
T
F
F
F
T
F
F
EXERCISE:
Use truth table to determine the argument form
pq
p ~q
pr
 r
is valid or invalid.
premises
conclusion
pq
p~q
pr
p
q
r
r
T
T
T
T
F
T
T
T
T
F
T
F
F
F
critical rows
T
F
T
T
T
T
T
T
F
F
T
T
F
F
F
T
T
T
T
T
T
F
T
F
T
T
T
F
F
F
T
F
T
T
T
F
F
F
F
T
T
F
The argument form is invalid
Page 20
Table of Contents:
  1. Recommended Books:Set of Integers, SYMBOLIC REPRESENTATION
  2. Truth Tables for:DE MORGANíS LAWS, TAUTOLOGY
  3. APPLYING LAWS OF LOGIC:TRANSLATING ENGLISH SENTENCES TO SYMBOLS
  4. BICONDITIONAL:LOGICAL EQUIVALENCE INVOLVING BICONDITIONAL
  5. BICONDITIONAL:ARGUMENT, VALID AND INVALID ARGUMENT
  6. BICONDITIONAL:TABULAR FORM, SUBSET, EQUAL SETS
  7. BICONDITIONAL:UNION, VENN DIAGRAM FOR UNION
  8. ORDERED PAIR:BINARY RELATION, BINARY RELATION
  9. REFLEXIVE RELATION:SYMMETRIC RELATION, TRANSITIVE RELATION
  10. REFLEXIVE RELATION:IRREFLEXIVE RELATION, ANTISYMMETRIC RELATION
  11. RELATIONS AND FUNCTIONS:FUNCTIONS AND NONFUNCTIONS
  12. INJECTIVE FUNCTION or ONE-TO-ONE FUNCTION:FUNCTION NOT ONTO
  13. SEQUENCE:ARITHMETIC SEQUENCE, GEOMETRIC SEQUENCE:
  14. SERIES:SUMMATION NOTATION, COMPUTING SUMMATIONS:
  15. Applications of Basic Mathematics Part 1:BASIC ARITHMETIC OPERATIONS
  16. Applications of Basic Mathematics Part 4:PERCENTAGE CHANGE
  17. Applications of Basic Mathematics Part 5:DECREASE IN RATE
  18. Applications of Basic Mathematics:NOTATIONS, ACCUMULATED VALUE
  19. Matrix and its dimension Types of matrix:TYPICAL APPLICATIONS
  20. MATRICES:Matrix Representation, ADDITION AND SUBTRACTION OF MATRICES
  21. RATIO AND PROPORTION MERCHANDISING:Punch recipe, PROPORTION
  22. WHAT IS STATISTICS?:CHARACTERISTICS OF THE SCIENCE OF STATISTICS
  23. WHAT IS STATISTICS?:COMPONENT BAR CHAR, MULTIPLE BAR CHART
  24. WHAT IS STATISTICS?:DESIRABLE PROPERTIES OF THE MODE, THE ARITHMETIC MEAN
  25. Median in Case of a Frequency Distribution of a Continuous Variable
  26. GEOMETRIC MEAN:HARMONIC MEAN, MID-QUARTILE RANGE
  27. GEOMETRIC MEAN:Number of Pupils, QUARTILE DEVIATION:
  28. GEOMETRIC MEAN:MEAN DEVIATION FOR GROUPED DATA
  29. COUNTING RULES:RULE OF PERMUTATION, RULE OF COMBINATION
  30. Definitions of Probability:MUTUALLY EXCLUSIVE EVENTS, Venn Diagram
  31. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:ADDITION LAW
  32. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:INDEPENDENT EVENTS