Sets
collection of distinct objects of same type or class of objects
A={1,2,3,4,5}
SET FORMATION
Builder form
Tabular form
actually listing its members
P = {a, b, c, d}.
If a set is defined by the properties which its elements must satisfy
T={x : x is even and less than 9}
STANDARD NOTATIONS
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x ∈ A x belongs to A or x is an element of set A.
x ∉ A x does not belong to set A.
φ Empty set.
U Universal set.
N The set of all natural numbers.
I The set of all integers.
I0 The set of all non-zero integers.
I+ The set of all + ve integers.
C, C0 The sets of all complex, non-zero complex numbers respectively.
Q, Q0, Q+ The sets of rational, non-zero rational, + ve rational numbers respectively.
R, R0, R+ The sets of real, non-zero real, +ve real number respectively.
VARIOUS TYPES OF SETS
Disjoint Sets
Family of Sets
Equality of Sets
Subset of a Set
Uncountable Set
Null Set or Empty Set
Countable Set
Power Set
Infinite Set
Universal Set
Finite Set
Comparable and Incomparable Sets
{ } enclose elements in set
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consists of specific number of different elements
consists of infinite number of different elements
if it is finite or denumerable se
is infinite and is not cardinally equivalent to N or is a non-denumerable set.
sets A and B are said to be equal and written as A = B if both have the same elements.
Two sets A and B are said to be disjoint if no element of A is in B and no element of B is in A.
If a set A contains elements which are itself sets then it is called family of sets or a set of sets
If every element of a set A is also an element of a set B then A is called subset of B and is written as A ⊆ B. B is called superset of A.
If A is subset of B and A ≠ B then A is said to be proper subset of B
Improper Subset. If A is subset of B and A = B, then A is said to be an improper subset of B
contains no element is called the null set or the empty set and is denoted by φ
power set of any given set A is the set of all subsets of A and is denoted by P(A). If A has n elements then P(A) has 2n elements
f all the sets under investigation are subsets of a fixed set U, then the set U is called universal set
Two sets A and B are comparable if one of them is subset of other,
A ⊆ B or B ⊆ A. If A ⊆ B and B ⊆ A, then A = B. Set φ is comparable to every set. Every set is comparable to the universal set U.
Two sets A and B are said to be incomparable if A ⊈ B and also B ⊈ A
there is at least one element in A not in B and vice-versa.
OPERATIONS ON SETS
Difference of Sets
Complement of a Set w.r.t. a Universal Set
Intersection of Sets
Symmetric Difference of Sets
Union of Sets
elements which belong to A or B or both
denoted by A ∪ B.
set of all those elements which belong to both A and B
denoted by A ∩ B.
set of all those elements which belong to A but do not belong to B
denoted by A – B
complement of a set A is a set of all those elements of the universal set which do not belong to A
denoted by Ac
the set containing all the elements that are in A or in B but not in both
denoted by A ⊕ B i.e
ALGEBRA OF SETS
Idempotent Laws
A ∪ A = A
A ∩ A = A
Associative Laws
(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)
Commutative Laws
A ∪ B = B ∪ A
A ∩ B = B ∩ A
Distributive Laws
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
De Morgan's Laws
(A ∪ B)c = Ac ∩ Bc
(A ∩ B)c = Ac ∪ Bc
Identity Laws
A ∪ φ = A
A ∩ U = A
A ∪ U = U
A ∩ φ = φ
Complement Laws
A ∪ Ac = U
A ∩ Ac = φ
Uc = φ
φc = U
Involution Law
(Ac)c = A
CARDINALITY OF A SET
The total numbers of unique elements in the set
Let P = {k, l, m, n}
The cardinality of the set P is 4
VENN DIAGRAMS
Venn diagram to represent the sets A, Ac and U
The Venn diagram for A ∪ B
Venn diagram for A ∩ B
Venn diagram for Ac ∩ Bc ∩ Cc
MULTISETS
A multiset is an unordered collection of elements, in which the multiplicity of an element may be one or more than one or zero
The multiplicity of an element is the number of times the element repeated in the multiset
ORDERED PAIRS
two elements designated 1st member and 2nd member written as (p, q).
CARTESIAN PRODUCT OF TWO SETS
P and Q in that order is the set of all ordered pairs whose first member belongs to the set P and second member belongs to set Q and is denoted by P × Q