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Elementary Mathematics

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MTH001 ­ Elementary Mathematics
LECTURE # 4
BICONDITIONAL
If p and q are statement variables, the biconditional of p and q is
"p if, and only if, q" and is denoted pq. if and only if abbreviated iff.
The double headed arrow " " is the biconditional operator.
TRUTH TABLE FOR
pq.
p q
p
q
T
T
T
T
F
F
F
T
F
F
F
T
EXAMPLES:
True or false?
1."1+1 = 3 if and only if earth is flat"
TRUE
2. "Sky is blue iff 1 = 0"
FALSE3. "Milk is white iff birds lay eggs"
TRUE
4. "33 is divisible by 4 if and only if horse has four legs"
FALSE
5. "x > 5 iff x2 > 25"
FALSE
pq (pq)(qp)
pq
pq
qp
(pq)(qp)
p
q
T
T
T
T
T
T
T
F
F
F
T
F
F
T
F
T
F
F
F
F
T
T
T
T
same truth values
REPHRASING BICONDITIONAL:
pq is also expressed as:
"p is necessary and sufficient for q"
"if p then q, and conversely"
"p is equivalent to q"
EXERCISE:
Rephrase the following propositions in the form "p if and only if q" in English.
1.If it is hot outside you buy an ice cream cone, and if you buy an ice cream
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MTH001 ­ Elementary Mathematics
cone it is hot outside.
Sol
You buy an ice cream cone if and only if it is hot outside.
2.For you to win the contest it is necessary and sufficient that you have the
only winning ticket.
Sol You win the contest if and only if you hold the only winning ticket.
3.If you read the news paper every day, you will be informed and conversely.
Sol You will be informed if and only if you read the news paper every day.4.It rains if it is a
weekend day, and it is a weekend day if it rains.
Sol It rains if and only if it is a weekend day.
5.The train runs late on exactly those days when I take it.
Sol The train runs late if and only if it is a day I take the train.
6.This number is divisible by 6 precisely when it is divisible by both 2 and 3.
Sol This number is divisible by 6 if and only if it is divisible by both 2 and 3.
TRUTH TABLE FOR
(pq) (~ q~ p)
pq
~ q~ p
(pq) (~ q~ p)
p
q
~q
~p
T
T
T
F
F
T
T
T
F
F
T
F
F
T
F
T
T
F
T
T
T
F
F
T
T
T
T
T
TRUTH TABLE FOR
(pq)(rq)
pq
rq
(pq)(rq)
p
q
r
T
T
T
T
T
T
T
T
F
T
F
F
T
F
T
F
F
T
T
F
F
F
T
F
F
T
T
F
T
F
F
T
F
F
F
T
F
F
T
T
F
F
F
F
F
T
T
T
TRUTH TABLE FOR
p ~r qr
Here p ~ r q r means (p (~ r)) (q r)
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MTH001 ­ Elementary Mathematics
p~r
qr
p ~r qr
p
q
r
~r
T
T
T
F
F
T
F
T
T
F
T
T
T
T
T
F
T
F
F
T
F
T
F
F
T
T
F
F
F
T
T
F
F
T
F
F
T
F
T
F
T
F
F
F
T
F
F
T
F
F
F
F
T
F
F
T
LOGICAL EQUIVALENCE
INVOLVING BICONDITIONAL
Show that ~pq and p~q are logically equivalent
~pq
p~q
p
q
~p
~q
T
T
F
F
F
F
T
F
F
T
T
T
F
T
T
F
T
T
F
F
T
T
F
F
same truth values
EXERCISE:
Show that ~(pq) and pq are logically equivalent
pq
~(pq)
pq
p
q
T
T
F
T
T
T
F
T
F
F
F
T
T
F
F
F
F
F
T
T
same truth values
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MTH001 ­ Elementary Mathematics
LAWS OF LOGIC:
pqqp
1.Commutative Law:
p q ~p q
2.Implication Laws:
~(p ~q)
(p q)r p (q r)
3.Exportation Law:
p q (p q)(q p)
4.Equivalence:
p q (p ~q) c
5.Reductio ad absurdum
APPLICATION:
Rewrite the statement forms without using the symbols or
1.p~qr
2. (pr)(q r)
SOLUTION
1.p~qr (p~q)r
order of operations
~(p~q) r
implication law
2.(pr)(q r) (~p r)(~q r) implication law
[(~p r) (~q r)] [(~q r) (~p r)]
equivalence of biconditional
[~(~p r) (~q r)] [~(~q r) (~p r)]
implication law
Rewrite the statement form ~p q r ~q to a logically equivalent form that uses
only ~ and
SOLUTION
STATEMENT
REASON
~p q r ~q
Given statement form
(~p q) (r ~q)
Order of operations
~[(~p q) ~ (r ~q)]
Implication law pq ~(p~ q)
~[~(p ~q) (~r q)]
De Morgan's law
Show that ~(pq) p is a tautology without using truth tables.
SOLUTIONSTATEMENT
REASON
~(pq) p
Given statement form
~[~(p ~q)] p
Implication law pq ~(p ~q)
(p ~q) p
Double negation law
~(p ~q) p
Implication law pq ~p q
(~p q) p
De Morgan's law
(q ~p) p
Commutative law of
∨≡ q (~p p)
Associative law of
∨≡ q t
Negation law
t
Universal bound law
EXERCISE:
Suppose that p and q are statements so that pq is false.
Find the truth values
of each of the following:
1.~ p q
2.p q
3.q p
SOLUTION
1.TRUE
2.TRUE
3.FALSE
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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