SET THEORY:
Set theory, branch of mathematics that
deals with the properties of well-defined collections of distinct objects,
which may or may not be of a mathematical nature, such as numbers or functions.
The sets are represented
with capital letters.
The objects are
called elements or
members of the set.
Elements or
members of the set.
Can be anything: numbers,
people, letters of the alphabet, other sets, and so on.
To indicate that an
object x is a member of a set A one
writes x ∊ A,
while x ∉ A indicates
that x is not a member of A.
Some Example of Sets
- A set of all positive integers
- A set of all the planets in the solar system
- A set of all the states in India
- A set of all the lowercase letters of the alphabet
Representation of a Set
Sets can be represented in two ways −
- Roster or Tabular Form
- Set Builder Notation
Roster or Tabular Form
The set is represented by listing all the elements comprising it.
The elements are enclosed within braces and separated by commas.
Example 1 − Set of vowels in English
alphabet, A={a,e,i,o,u}
Example 2 − Set of odd numbers less
than 10, B={1,3,5,7,9}
Set Builder
Notation
The set is defined by specifying a
property that elements of the set have in common. The set is described as A={x:p(x)
Example 1 − The set {a,e,i,o,u} is written as −
A={x:x is a vowel in English alphabet}
Example 2 − The set {1,3,5,7,9} is written
as −
B={x:1≤x<10 and (x%2)≠0}
If an element x is a member of any set S,
it is denoted by x∈S and if an element y is not a member of set S, it is denoted
by y∉S
Example − If S=S={1,1.2,1.7,2},1∈S but 1.5∉S
Some Important Sets
N − the set of all natural numbers
= {1,2,3,4,.....}
Z – the set of all integers= {.....,−3,−2,−1,0,1,2,3,.....}
Z+ − the set of all positive integers
Q − the set of all rational numbers
R − the set of all real numbers
W − the set of all whole numbers
Cardinality
of a Set
Cardinality of a set S, denoted by |S|, is the number of
elements of the set. The number is also referred as the cardinal number. If a
set has an infinite number of elements, its cardinality is ∞.
Example − |{1,4,3,5}|=4,
|{1,2,3,4,5,…}|=∞
If there are two sets X and Y,
·
|X|=|Y| denotes two sets X and Y having same cardinality. It
occurs when the number of elements in X is exactly equal to the number of
elements in Y. In this case, there exists a bijective function ‘f’ from X to Y.
·
|X|≤|Y| denotes that set X’s cardinality is less than or
equal to set Y’s cardinality. It occurs when number of elements in X is less
than or equal to that of Y. Here, there exists an injective function ‘f’ from X
to Y.
·
|X|<|Y| denotes that set X’s cardinality is less than set Y’s
cardinality. It occurs when number of elements in X is less than that of Y.
Here, the function ‘f’ from X to Y is injective function but not bijective.
·
If |X|≤|Y| and |X|≥|Y| then |X|=|Y|. The sets X and Y are commonly referred as equivalent
sets.
Finite Set
A set which contains a definite number of elements is called a
finite set.
Example − S={x|x∈N and 70>x>50}
Infinite Set
A set which contains infinite number of elements is called an
infinite set.
Example − S={x|x∈N and x>10}
Subset
A set X is a subset of set Y (Written
as X⊆Y) if every element of X is
an element of set Y.
Example 1 − Let, X={1,2,3,4,5,6} and Y={1,2}. Here set Y is a subset of set X as all the elements of
set Y is in set X. Hence, we can write Y⊆X.
Example 2 − Let, X={1,2,3} and Y={1,2,3}. Here set Y is a subset (Not a proper subset) of set X as
all the elements of set Y is in set X. Hence, we can write Y⊆X
Proper
Subset
The term “proper subset” can be defined as
“subset of but not equal to”. A Set X is a proper subset of set Y (Written
as X⊂Y) if every element of X is
an element of set Y and |X|<|Y|.
Example − Let, X={1,2,3,4,5,6} and Y={1,2} Here set Y⊂X since all elements
in Y are contained in X too and X has
at least one element is more than set Y.
Universal
Set
It is a collection of all elements in a
particular context or application. All the sets in that context or application
are essentially subsets of this universal set. Universal sets are represented
as U.
Example − We may define U as the set of all
animals on earth. In this case, set of all mammals is a subset of U, set of all fishes is a subset of U, set of all insects is a subset of U, and so on.
Empty Set or
Null Set
An empty set contains no elements. It is
denoted by ∅. As the number of elements in an empty set is finite, empty set
is a finite set. The cardinality of empty set or null set is zero.
Example −S={x|x∈N and 7<x<8}=∅
Singleton
Set or Unit Set
Singleton set or unit set contains only
one element. A singleton set is denoted by {s}.
Example − S={x|x∈N, 7<x<9}
Equal Set
If two sets contain the same elements they are said to be equal.
Example −
If A={1,2,6} and B={6,1,2} they
are equal as every element of set A is an element of set B and every element of
set B is an element of set A.
Equivalent Set
If the
cardinalities of two sets are same, they are called equivalent sets.
Example −
If A={1,2,6} and B={16,17,22} they
are equivalent as cardinality of A is equal to the cardinality of B. i.e. |A|=|B|=3
Overlapping Set
Two sets
that have at least one common element are called overlapping sets.
In case
of overlapping sets −
·
n(A∪B)=n(A)+n(B)−n(A∩B)
·
n(A∪B)=n(A−B)+n(B−A)+n(A∩B)
·
n(A)=n(A−B)+n(A∩B)
·
n(B)=n(B−A)+n(A∩B)
Example −
Let, A={1,2,6} and B={6,12,42}
There is a common element ‘6’, hence these sets are overlapping sets.
Disjoint Set
Two sets
A and B are called disjoint sets if they do not have even one element in
common. Therefore, disjoint sets have the following properties −
·
n(A∩B)=∅
·
n(A∪B)=n(A)+n(B)
Example −
Let, A={1,2,6}and B={7,9,14},
there is not a single common element, hence these sets are overlapping sets.
Set Operations
Set
Operations include Set Union, Set Intersection, Set Difference, Complement of
Set, and Cartesian Product.
Set Union
The union of sets A and B (denoted by A∪BA∪B) is
the set of elements which are in A, in B, or in both A and B. Hence, A∪B={x|x∈A OR x∈B}A∪B={x|x∈A OR x∈B}.
Example −If A={10,11,12,13}A={10,11,12,13} and B= {13,14,15}{13,14,15}, then A∪B={10,11,12,13,14,15}A∪B={10,11,12,13,14,15}. (The common element occurs only once)
Set Intersection
The intersection of sets A and B (denoted by A∩BA∩B)
is the set of elements which are in both A and B. Hence, A∩B={x|x∈A AND x∈B}A∩B={x|x∈A AND x∈B}.
Example −If A={11,12,13}A={11,12,13} and B={13,14,15}B={13,14,15}, then A∩B={13}A∩B={13}.
Set Difference/ Relative Complement
The set difference of sets A and B (denoted by A–BA–B)
is the set of elements which are only in A but not in B. Hence, A−B={x|x∈A AND x∉B}A−B={x|x∈A AND x∉B}.
Example −
If A={10,11,12,13}A={10,11,12,13} and B={13,14,15}B={13,14,15}, then (A−B)={10,11,12}(A−B)={10,11,12} and (B−A)={14,15}(B−A)={14,15}. Here, we can see (A−B)≠(B−A)(A−B)≠(B−A)
Complement of a Set:
The complement of a set A (denoted by A′A′) is
the set of elements which are not in set A. Hence, A′={x|x∉A}
More specifically, A′=(U−A)A′=(U−A) where UU is a universal set which contains all objects.
Example − If A={x|x belongs to set of odd integers} then A′={y|y does not belong to set off odd integers}
Cartesian Product / Cross Product:
The Cartesian product of n number of sets A1,A2,…An denoted as A1×A2⋯×An can be defined as all possible ordered pairs (x1,x2,…xn)(x1,x2,…xn)where x1∈A1,x2∈A2,…xn∈A
Example − If we
take two sets A={a,b} and B={1,2}B={1,2},
The
Cartesian product of A and B is written as − A×B={(a,1),(a,2),(b,1),(b,2)}
The
Cartesian product of B and A is written as − B×A={(1,a),(1,b),(2,a),(2,b)}
Power Set
Power set of a set S is the set of all subsets of S
including the empty set. The cardinality of a power set of a set S of
cardinality n is 2n.
Power set is denoted as P(S).
Example
−
For a set S={a,b,c,d} let
us calculate the subsets −
·
Subsets
with 0 elements − {∅} (the
empty set)
·
Subsets
with 1 element − {a},{b},{c},{d}
·
Subsets
with 2 elements − {a,b},{a,c},{a,d},{b,c},{b,d},{c,d}
·
Subsets
with 3 elements − {a,b,c},{a,b,d},{a,c,d},{b,c,d}
·
Subsets
with 4 elements − {a,b,c,d}
Hence, P(S)=
{{∅},{a},{b},{c},{d},{a,b},{a,c},{a,d},{b,c},{b,d},{c,d},{a,b,c},{a,b,d},{a,c,d},{b,c,d},{a,b,c,d}}
|P(S)|=24=16
Note − The power set of an empty set is also an empty set.
|P({∅})|=20=1
Partitioning of a Set:
Partition of a set, say S, is a collection
of n disjoint subsets, say P1,P2,…Pn that
satisfies the following three conditions −
·
Pi does
not contain the empty set.
[Pi≠{∅} for all 0<i≤n]
·
The
union of the subsets must equal the entire original set.
[P1∪P2∪⋯∪Pn=S]
·
The intersection
of any two distinct sets is empty.
[Pa∩Pb={∅}, for a≠b where n≥a,b≥0]
Example
Let S={a,b,c,d,e,f,g,h}
One probable partitioning is {a},{b,c,d},{e,f,g,h}
Another probable partitioning is {a,b},{c,d},{e,f,g,h}.
Laws of Algebra of Sets:
For three sets A, B and C;
(i) Idempotent Law
(a) A∪ A =A (b) A ∩
A = A
(ii) Identity Law
(a) A∪ φ
= A (b) A ∩ φ = φ
(iii) Commutative Law
(a) A∪ B= B ∪A (b) A∩
B= B∩A
(iv) Associative Law
(a) A∪ (B∪C) = (A∪B)∪C b) A∩(B∩C)= (A∩B)∩C
(v) Distributive Law
(a) A∪ (B ∩C)
=(A∪
B) ∩ (A∪
C) (b) A∩
(B ∪C)
=(A∩ B) ∪ (A∩
C)
Assignment-2
1.Draw
the digraph of R and list the in-degree and out-degree of all
Vertices.
2.Let
A={1,2,3,4} and R={(1,1),(1,2),(2,1),(2,2),(3,4),(4,3),(3,3),(4,4)} be relation
on A. Verify whether R is an equivalence relation.
3.
Explain about following properties on relations with examples:
i) Reflexive
ii) Transitive iii) Symmetric
4. Let A= {1, 2, 3, 4, 6, 12}. On A, define the relation R by
aRb if and only if a divides b. Prove that R is a partial order relation on A.
Draw the Hasse diagram for this relation.
5. If
A={1,2,3,4} and R,S are relations on A defined by R={(1,2), (1,3), (2,4),
(4,4)} S={(1,1), (1,2), (1,3), (1,4),
(2,3), (2,4)} find RoS, SoR,R2,S2
and write down their matrices
Tutorial-III
1.Let
A={1,2,3,4} and R={(1,1),(1,2),(2,1),(2,2),(3,4),(4,3),(3,3),(4,4)} be relation
on A. Verify whether R is an equivalence relation.
2.
Explain about following properties on relations with examples:
i) Reflexive
ii) Transitive iii) Symmetric
3. Let A={1, 2, 3, 4, 6, 12}. On A, define the relation R by
aRb if and only if a divides b. Prove that R is a partial order relation on A.
Draw the Hasse diagram for this relation.
Tutorial-IV
1. Assume A={1,2,3,4} and f and g be functions
from A to A given by f={(1,4),(2,1),(3,2),(4,3)} and
g={(1,2),(2,3),(3,4),(4,1)}. Prove that f and g are inverse of each other
2. If
A={1,2,3,4} and R is a relation on A defined by R={(1,2), (1,3),(2,4), (3,2), (3,3),
(3,4)}, find M(R) , M(R2),
M(R3). Verify that M(R2) = [M(R)]2 and M(R3)=[M(R)]3.
3. Consider the functions f and g defined by f(x)=x3 and g(x)=x2+1, for all x ∈R.
Find g o f, f o g, f2 and g2
4. If U={1,2,3,4,5,6,7,8,9}, A={1,2,4,6,8} and B={2,4,5,9}
then compute the following
i) A1 ii) A1 U B1 iii) (A∩B)1 iv) AΔB
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