Tuesday, 26 February 2013

Chapter 1


1.     PROPOSITIONAL LOGIC

Proposition or statements, are also known as simple sentences makes up the basic units for propositional logic. When two or more simple sentences are related through connectives, it forms a more complicated proposition called compound sentence.
A statement / propositions must always be able to be validated as True (T) or False (F).

For example:
i) All fathers are men.
This statement is true because only man can be father.
ii) All men are father.
This statement is false because only those men who are married and have kids are fathers. Single men are not father.
iii) I like Monday.
This sentence is neither true nor false as it does not include a fact. The notion of ‘liking Monday’ is entirely depending on the particular individual. Some may like it. Some may not.

Sentence which are neither true nor false (not containing any facts) are not propositions.


TRUTH VALUE

The truth value of a true proposition / statement is always T. While the truth value of a false proposition / statement is always F.


PROPOSITIONAL VARIABLE

Propositional variables are used to stand for unspecified statements. Means that each variable assigned may or may not have connection between them.

For example:
A = She likes dancing.
B = She likes camping.
C = She likes outdoor activities.
Each variable is assigned to a statement.
Now, let’s see how we can relate these variables.


Conjunction [AND]

The conjunction of two statements A and B is only true when both statements are true. And the conjunction of the two statements is false if one of them is false.
Scenario: You’re grading curriculum achievement for Grace. She will only get A for her curriculum if she participated in more than 1 activity. So, the only way for her to score A in her curriculum is:

A
B
(A & B)
T
T
F
F
T
F
T
F
T (She likes both dancing and camping.)
F (She likes dancing but not camping.)
F (She doesn’t like dancing but fond of camping.)
F (She doesn’t like both dancing and camping.)

*The conjunction of the two statements requires that she likes both activities (dancing and camping) for the compound statement to be true (she score A for her curriculum activities). The compound statement is false if one or both of the simple sentences is false. Conjunction using the operator ‘&’ is the equivalent of joining statements together with ‘and’ in English.


Disjunction [OR]

The disjunction of two statements is true even if only one of the two statements is true. It will only be false if and only if both statements are false.
Scenario: You’re trying to impress a girl by finding conversation topics you two have in common. You’re a great dancer and camper having lots of experience in both activities. You’ll succeed in getting her attention if:

A
B
(A & B)
T
T
F
F
T
F
T
F
T (She likes both dancing and camping.)
F
(She likes dancing but not camping.)
F
(She doesn’t like dancing but fond of camping.)
F (She doesn’t like both dancing and camping.)

*The disjunction of two statements are true (you succeeded in finding a topic in common) if both / one of the two statements is true (that she is either interested in dancing or camping). You will only fail (disjunction is a false statement) if she is not interested in both dancing and camping (both simple sentence that makes up the disjunction are false). This is equivalent of joining statements together with the word ‘or’ in English.


Material Implication[If… Then…]

With A as antecedent, and B as consequence, statement / incident A catalyzes the event / statement B which then determines the overall truth value of the statement.  The compound statement formed will only be false if A is true but B is false. The statement will be true if both are true or false, and either A is false or B is true.
Take:
A = It’s cloudy outside.
B = I bring umbrella.
For (A→B), I bring umbrella because it’s cloudy outside.
Because of the truth of the statement A (It’s cloudy outside), that catalyses the event B (me bringing umbrella).

A
B
(A → B)
T
T
F
F
T
F
T
F
T (It’s raining so I brought umbrella.)
F
(It’s raining but I didn’t bring umbrella?)
T
(It’s not raining but I brought umbrella?)
T
(It’s not raining so I didn’t bring umbrella.)

This is equivalent of joining statements together with the word ‘if… then / so…’ in English.


Material Equivalence[If and only if]

Statements formed by material equivalence are counted as true if both A and B have the same truth value, and will be regarded as false if one of the statements is false. Hence:

A
B
(A ↔ B)
T
T
F
F
T
F
T
F
T
F
F
T

Since this statement requires both statements to have the same truth value, this is akin to joining statements together with the word ‘… if and only if… ’ in English.



2.     PROPOSITION EQUIVALENCE

Definition of :
·         Tautology
A compound proposition that is always true.

·         Contradiction
A compound proposition that is always false

·         Contingency
A compound proposition that is neither a tautology nor a contradiction.



LOGICAL EQUIVALENCES

Compound proposition that have the same logical content. Means, compound propositions p and q are called logically equivalent if p ↔ q is a tautology.


Example of Tautology and Contradiction

p
q
p  -p
 -p
T
T
T
F
F
F
T
F

*p -q is always true
*p -q is always false


Logical equivalences tables

Name
Equivalence
Identity Laws
 T ≡ p
 F ≡ p
Domination Laws
 T ≡ T
 F ≡ F
Idempotent Laws
 p ≡ p
 p ≡ p
Negation Laws
¬p ≡ T
¬p ≡ F
Double Negation Law
¬(¬p) ≡ p
Commutative Law
 q ≡ q  p
 q ≡ q  p
Absorption Law
 (p  q) ≡ p
 (p  q) ≡ p
De Morgan’s Laws
¬(p  q) ≡¬p ¬q
¬(p  q) ≡¬p ¬q
Associative Laws
(p  q)  r ≡ p  (q  r)
(p  q)  r ≡ p  (q  r)
Distributive laws
 (q  r) ≡ (p  q)  (p  r)
 (q  r) ≡ (p  q)  (p  r)

*T = Compound proposition is always true
*F = Compound proposition is always false



Logical Equivalences involving Conditional Statements

p → q ≡ ¬p  q
p → q ≡ ¬q →¬p
 q ≡¬p → q
 q ≡¬(p →¬q)
¬(p → q) ≡ p ¬q
(p → q)  (p → r) ≡ p → (q  r)
(p → r)  (q → r) ≡ (p  q) → r
(p → q)  (p → r) ≡ p → (q  r)
(p → q)  (p → r) ≡ p → (q  r)


Logical Equivalences involving Bi-conditional Statements

p ↔ q ≡ ¬p ↔¬q
¬(p ↔ q) ≡ p ↔¬q
p ↔ q ≡ (p  q)  (¬p ¬q)
p ↔ q ≡ (p → q)  (q → p)


Logical Equivalent that are logical equivalents.
 

Double Negation Law
¬ ( ¬ p ) ≡ p

p
-p
-(-p)
T
F
T
F
T
F


            Commutative Law
            p  q ≡ q  p



p
q
 q
 p
T
T
T
T
T
F
T
T
F
T
T
T
F
F
F
F


                                                                     
Associative Law
(p  q)  r ≡ p  (q  r


p
q
r
(p  q)
(p  q)  r
(q  r)
 (q  r)
T
T
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
F
F
T
T
T
T
T
F
F
F
T
T
T
T
F
F
T
F
T
T
T
F
F
F
F
F
F
F



Logical Equivalent involving Conditional Statement that are logical equivalents.
 
                                                                    ¬(p → q) ≡ p ¬q


p
q
¬(p → q)
¬q
¬q
T
T
F
F
F
T
F
T
T
T
F
T
F
F
F
F
F
F
T
F


(p→q)  (p → r) ≡ p → (q  r)
 

p
q
r
(p → q)
(p → r)
(p → q)  (p → r)
(q  r)
p → (q  r)
T
T
T
T
T
T
T
T
T
T
F
T
F
T
T
T
T
F
T
F
T
T
T
T
T
F
F
F
F
F
F
F
F
T
T
T
T
T
T
T
F
T
F
T
T
T
T
T
F
F
T
T
T
T
T
T
F
F
F
T
T
T
F
T


Logical Equivalent involving Bi-conditional

p ↔ q ≡ (p → q)  (q → p) 

P
q
p ↔ q
p → q
q → p
(p → q)  (q → p)
T
T
T
T
T
T
T
F
F
F
T
F
F
T
F
T
F
F
F
F
T
T
T
T





 3.     PREDICATES AND QUANTIFIERS

PREDICATES

In some situations, a statement can be composed of two parts, the variable and the predicates.

Take for an example.
Statement: The cost of an apple is RM x.
(In this case, the cost of the apple is a predicate, while x is the variable which represents the price.)
The above statement can also be expressed as propositional function P at the value of x, which is  P(x), Where P[the cost of an apple] is affected by x[the price].
Note: A predicate is not a proposition until it is bound by the variable. In this case, the predicate is not a proposition until variable is declared on it. P itself is not a proposition until it is bound by x, forming P(x).
Take P(x): x+5>10, it has no value until x is substituted with a finite number or value.
When x is assigned a value, a proposition is formed.
P (4) is false. [4+5<10]
P (5) is false. [5+5=10]
P (6) is true. [6+5>10]
P (7) is true. [7+5>10]

 P (v) ¬P (0) is not a proposition as the value of ‘v’ is not assigned.
However, P (3) ¬P (0) is a proposition which is true.


QUANTIFIER

Universal Quantifier
is used to represent ‘for all’.
x P (x) is expressed as “For all (values of) x, P(x) is…” or x1 x2 x3 …… xn
The variable x is bound by the universal quantifier that results in forming of a proposition.


Existential Quantifier
is used to represent “for some state / value of “
P (x) is true for some value of x in the universal set.
x P(x) is also expressed as “There’s an x such that P(x),”, “For at least one value of x, P(x)…”


UNIQUENESS QUANTIFIER
! Is used to represent “for one and only one x”
P (x) is only true for one value of x.
! x P (x) is also expressed as “The is one and only one x such that P (x)”, “The is a unique x such that P (x)…”.