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Introduction To Languages
And
Theory Of computation
Basic Mathematical Objects
Sets
What is a Set?
Set is a group of elements, having a property that characterizes those elements.
How to specify a Set?
One way is to enumerate the elements completely ( all the elements belonging to
set are explicitly given) and give a property that characterizes the elements of the Set.
Example:
A = {1,2,3,4,5}
Other way is to enumerate the elements in a way that makes clear what the
remaining elements are and to give a property that characterizes the elements of the Set.
Example:
B = {x/x is a positive integer less than or equal to 5}
C = {1,2,3,4,5,6,……..} ( this is an infinite set)
Add "Recursive definition of sets" with an example.
How to play around with Sets? What do you mean by this ?
x E B implies that x is an element of set B. Using this notation we can specify the
set {1,2,3,4,5} by writing
Z = {x  A \ x <= 5} { x | x  A and x <= 5 }
which is read as “ the set of elements ‘x’ in ‘A’such that ‘x’<5”.
Subset:
If A and B are two sets,
A is a subset of B, if every element of A is an element of B.
Example: A = {1,2,3}
B = {x/x is a positive integer}
Here we can think of some better way of specifying the set B just by replacing the
English like sentence “x is a positive integer less than 100” in a more mathematically
simpler way. We can achieve it by assuming ‘N’ as all positive integers.
We can specify set B as,
B = {x / x E N} If you use N, then you might as well say A N .
1,2,3 are all clearly one of the elements in set B as they satisfy the property
that characterizes the elements of set B. (they all are positive integers)
A is a subset of B is represented as AB.
If A is a subset of B and B is a subset of A then AB. Also, A is subset and equal to B
represented as AB and BA.
Compliment: Complement
If A is a set, then the compliment of A is the set A’ consisting of all elements contained
in the universal set and not in A.
Universal Set:
A set U that contains all the elements we might ever consider. Put this before
Complement
Then the complement of A, denoted by A’, can be represented as
A’={x  U | x A}
Where  “is not an element of”.
E.g.: if U={1,2,3….}
A= {1,2,3} then A’={4,5,6….}
The operations that can be performed on sets are:
1. Union: If A and B are two sets, then A union B is a set that contains all the
elements only in A, only in B and in both A and B which can be represented as
AB.
E.g.: A={1,2,3} B={3,4,5} AB={1,2,3,4,5}
2. Difference: If A and B are two sets, then A Difference B is a set that contains the
elements only in A and the elements only in B but not the elements that are in
both A and B which can be represented as A-B.
E.g.: A={1,2,3} B={3,4,5} A-B={1,2,4,5}
3. Intersection: If A and B are two sets, then A intersection B is a set that contains
only the elements in both A and B which can be represented as AB.
E.g.: A={1,2,3} B={3,4,5} AB={3}.
Disjoint sets: A and B are said to be disjoint sets if they contain no elements in common
i.e. AB= where  is the Empty set that is the set with no elements.
Add Cartesian product, general union and intersection (i.e. union/intersection of
three or more sets -- pp. 8 - 9 of textbook).
Here is a list of some standard set identities:
A, B, C represent arbitrary sets and  is the empty set and U is the Universal Set.
The Commutative laws:
AB=BA
AB=BA
The Associative laws:
A(BC)=(AB)C
A(BC)=AB) C
The Distributive laws:
A(BC)=(AB)  (AC)
A(BC)=(AB) (AC)
The Idempotent laws:
AA=A
AA=A
The Absorptive laws:
A (AB)=A
A(AB)=A
The De Morgan laws:
(AB)’=A’B’
(AB)’=A’B’
Other laws involving Complements:
(A’)’=A
AA’=
AA’=U
Other laws involving the empty set:
A=A
A=
Other laws involving the Universal Set:
AU=U
AU=A
Will include proof for one of the Demorgan’s laws in the Site.
Logic
1. Proposition and Logical Connections: A “Proposition” can be defined as a
declarative statement with sufficient meaning, objective and a definitive truthvalue: true/false.
Connective: Two or more propositions can be combined together to make
compound propositions with the help of logical connectives to study
express/represent the logical arguments.
The following are the logical connectives used commonly:
a. Conjunction:
The logical conjunction is understood in the same way as commonly used “and”.
The compound proposition truth-value is true iff all the propositions hold true. It
is represented as “^”.
Truth table for two individual propositions p and q with conjunction is
The truth value of the conjunction of two propositions p and q is defined
using the truth table as :
p
q
p^q
T
T
T
T
F
F
F
T
F
F
F
F
b. Disjunction: This is logical “or” read as either true value of the individual
propositions.
Truth table is see conjunction
p
q
pVq
T
T
T
T
F
T
F
T
T
F
F
F
c. Negation: This is the logical “Negation” read as for p “not p”.
Truth table is see conjunction
p
p
T
F
F
T
d. Conditional: This is used to define as “a proposition holds true” iff another
proposition as true i.e. p->q is read as “if p, then q”.
Truth table is see conjunction
p
q
p->q
T
T
F
T
F
F
F
T
T
F
F
T
e. Biconditional: A proposition (p->q)^(q->p) can be abbreviated using
biconditional conjunction <-> as p<->q and is read as “p only if q” and “p if q”.
"p if and only if q"
f. Tautology: A compound proposition, which is true in every case.
E.g.: pV q p V p
g. Contradiction: This is the opposite of tautology, which is false in every case.
E.g.: p^q p^p
Logical implication and equivalence: If the value of Q q is true in every case in,
which p is true then p is said to logically imply q, which is represented as p=>q.
If p and q have same truth-value in each case then both are said to be logically
Equivalent represented as p<=>q.
Logical quantifies and quantified statements: Quantifiers are used to make a
proposition about a domain individual objects. The proposition variable is bound
to a particular
domain and not a free variable. Such statements with quantifiers is called a
“quantified statement”.
E.g.: Existential quantifiers 
Universal quantifier 
Need more on Predicate Logic. See my notes I gave you and also my Web
pages on predicate logic.
Need an entry on "Relations" here.
Functions:
If f is a function and x is an element of its domain, the element of the co-domain
associated with x is written as f(x). Move this down below the next paragraph.
A function associates with, or assigns to each element of one set to a single element in the
other set. The former set is called the “domain” and the later is called “Co-domain”.
It A function is represented as follows:
f : A  B where f is the function, A is the domain and B is the co-domain.
Types of functions:
1. Onto functions: If A and B are two sets, such that f: A  B and every element in
B is associated with an element in A then f is said to be an onto, surjective or
surjection i.e. f(A)=B. Define this notation if you want to use it.
1
2
3
4
5
a
b
c
2. One-to-One functions: If A and B are two sets, such that f: A  B and if no
single element of B is associated with more than one element in A, then the
function f is said to be One-to-One or injective or an injection.
1
2
3
4
a
b
c
4 must be mapped to something in B. Otherwise you don't have a function.
3. Bijection: A function is said to be bijection if it is both Onto and One-to-One.
1
2
3
a
b
c
Composition of functions:
If f: A  B and g: BC then, for any element xA, the function h: AC defined by
h(x)=g(f(x)) is called the composition of g and f and is written as h=g o f.
Inverse of a function:
If f: AB is a bijection, then for any yB and xA such that y=f(x), the inverse of f
denoted by f-1 is defined as x=f-1(y).