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Recommended Books:Set of Integers, SYMBOLIC REPRESENTATION

Truth Tables for:DE MORGAN’S LAWS, TAUTOLOGY >>
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MTH001 ­ Elementary Mathematics
LECTURE # 1
1.Recommended Books:
1.Discrete Mathematics with Applications (second edition) by Susanna S. Epp
MAIN TOPICS:
1. Logic
2. Sets & Operations on sets
3. Relations & Their Properties
4. Functions
5. Sequences & Series
Set of Integers:
·
·
·
·
·
·
3
-2
-1
0
1
2
Set of Real Numbers:
·
·
·
·
·
·
·
-3
-2
-1
0
1
2
What is Discrete Mathematics?:
Discrete Mathematics concerns processes that consist of a sequence of individual steps.
LOGIC:
Logic is the study of the principles and methods that distinguishes between a
valid and an invalid argument.
SIMPLE STATEMENT:
A statement is a declarative sentence that is either true or false but not both.
A statement is also referred to as a proposition
Example: 2+2 = 4, It is Sunday today
If a proposition is true, we say that it has a truth value of "true".
If a proposition is false, its truth value is "false".
The truth values "true" and "false" are, respectively, denoted by the letters T and F.
EXAMPLES:
Not Propisitions
1.Grass is green.
·
Close the door.
2.4 + 2 = 6
2.4 + 2 = 7
·
x is greater than 2.
3.There are four fingers in a hand.
·
He is very rich
are propositions
are not propositions.
Rule:
If the sentence is preceded by other sentences that make the pronoun or variable reference
clear, then the sentence is a statement.
Example
Example:
Bill Gates is an American
x=1
He is very rich
x>2
He is very rich is a statement with truth-value
x > 2 is a statement with truth-value
TRUE.
FALSE.
UNDERSTANDING STATEMENTS:
1.x + 2 is positive.
Not a statement
2.May I come in?
Not a statement
3.Logic is interesting.
A statement
4.It is hot today.
A statement
5.-1 > 0
A statement
6.x + y = 12
Not a statement
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MTH001 ­ Elementary Mathematics
COMPOUND STATEMENT:
Simple statements could be used to build a compound statement.
EXAMPLES:
LOGICAL CONNECTIVES
1.  "3 + 2 = 5" and "Lahore is a city in Pakistan"
2.  "The grass is green" or " It is hot today"
3.  "Discrete Mathematics is not difficult to me"
AND, OR, NOT are called LOGICAL CONNECTIVES.
SYMBOLIC REPRESENTATION:
Statements are symbolically represented by letters such as p, q, r,...
EXAMPLES:
p = "Islamabad is the capital of Pakistan"
q = "17 is divisible by 3"
CONNECTIVE
MEANINGS
SYMBOL
CALLED
Negation
not
~
Tilde
Conjunction
and
Hat
Disjunction
or
Vel
Conditional
if...then...
Arrow
Biconditional
if and only if
Double arrow
EXAMPLES:
p = "Islamabad is the capital of Pakistan"
q = "17 is divisible by 3"
p q = "Islamabad is the capital of Pakistan and 17 is divisible by 3"
p q = "Islamabad is the capital of Pakistan or 17 is divisible by 3"
~p = "It is not the case that Islamabad is the capital of Pakistan" or simply
"Islamabad is not the capital of Pakistan"
TRANSLATING FROM ENGLISH TO SYMBOLS:
Let p = "It is hot", and q = " It is sunny"
SENTENCE
SYMBOLIC FORM
1.It is not hot.
~p
p q
2.It is hot and sunny.
pq
3.It is hot or sunny.
~ p q
4.It is not hot but sunny.
~p~q
5.It is neither hot nor sunny.
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MTH001 ­ Elementary Mathematics
EXAMPLE:
Let
h = "Zia is healthy"
w = "Zia is wealthy"
s = "Zia is wise"
Translate the compound statements to symbolic form:
(h w) (~s)
1.Zia is healthy and wealthy but not wise.
~w (h s)
2.Zia is not wealthy but he is healthy and wise.
~h ~w ~s
3.Zia is neither healthy, wealthy nor wise.
TRANSLATING FROM SYMBOLS TO ENGLISH:
Let
m = "Ali is good in Mathematics"
c = "Ali is a Computer Science student"
Translate the following statement forms into plain English:
1.~ c
Ali is not a Computer Science student
2.c m
Ali is a Computer Science student or good in Maths.
3.m ~c
Ali is good in Maths but not a Computer Science student
A convenient method for analyzing a compound statement is to make a truth
table for it.
A truth table specifies the truth value of a compound proposition for all
possible truth values of its constituent propositions.
NEGATION (~):
If p is a statement variable, then negation of p, "not p", is denoted as "~p"
It has opposite truth value from p i.e.,
if p is true, ~p is false; if p is false, ~p is true.
TRUTH TABLE FOR
~p
p
~p
T
F
F
T
CONJUNCTION ():
If p and q are statements, then the conjunction of p and q is "p and q", denoted as
"p q".
It is true when, and only when, both p and q are true. If either p or q is false, or
if both are false, pq is false.
TRUTH TABLE FOR
pq
pq
p
q
T
T
T
T
F
F
F
T
F
F
F
F
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MTH001 ­ Elementary Mathematics
DISJUNCTION ()
or INCLUSIVE OR
If p & q are statements, then the disjunction of p and q is "p or q", denoted as
"p q".It is true when at least one of p or q is true and is false only when both
p and q are false.
TRUTH TABLE FOR
pq
pq
p
q
T
T
T
T
F
T
F
T
T
F
F
F
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MTH001 ­ Elementary Mathematics
Note that in the table T is only in that row where both p and q have T
and all other values are F. Thus for finding out the truth values for the conjunction of two
statements we will only first search out where the
both statements are true and write down the T in the corresponding row
in the column of p q and in all other rows we will write F in the
column of p q.
DISJUNCTION () or INCLUSIVE OR
If p and q are statements, then the disjunction of p and q is "p or q", denoted as "p q"
It is true when at least one of p or q is true and is false only when both p and q are false.
Note that in the table F is only in that row where both p and q have F and all other values
are T. Thus for finding out the truth values for the disjunction of two statements we will only
first search out where the both statements are false and write down the F in the
corresponding row in the column of p q and in all other rows we will write T in the column
of p q.
Remark:
Note that for Conjunction of two statements we find the T in both the
statements, But in disjunction we find F in both the statements. In other words we will
fill T first in the column of conjunction and F in the column of disjunction.
SUMMARY
1. What is a statement?
2. How a compound statement is formed.
3. Logical connectives (negation, conjunction, disjunction).
4. How to construct a truth table for a statement form.
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Table of Contents:
  1. Recommended Books:Set of Integers, SYMBOLIC REPRESENTATION
  2. Truth Tables for:DE MORGAN’S 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