Sets & Basic Operations: Set Theory - 01
Sets & Basic Operations: Set Theory - 01
Sets & Basic Operations: Set Theory - 01
A set is normally denoted by a capital letter, such as A, X etc., while the elements might be denoted generically by
lowercase letters like a, b etc. The elements of a set are enclosed within curly braces '{' and '}', and are separated
from each other by commas.
Sets can be specified in two ways:
1.
Tabular form:
The set is defined by stating the properties which characterize the elements of the set. e.g., B={x:x
odd integer, x>0}
is an
The above reads as is a set of such that is an odd integer and . x stands for any element of the
set, the colon is read as such that and the comma as and. In explicit form, the above set
is .
The symbolic way of writing the statement that an element is an element of set or, equivalently,
belongs to set , ispA
When specifying two elements, like and , we writep,qA
The statement does not belong to is written asaA
The set
is a vowel}
Two sets and are equal if both have the same elements. This is denoted as . If they areunequal, it is
expressed as AB
Note that changing the arrangement of the elements of a set does not change the set . Also, repetition of elements
of a set does not change the set.
Three sets are defined as follows:
, and
,
Subsets
If every element of a set is also an element of set , then is called a subset of . This is also expressed
as: is contained in , or contains . This is expressed symbolically asAB
Inversely, B is called a superset of . Symbolically, this is expressed asBA
The statement A is not a subset of B implies that at least one element of A does not belong to B. This is
expressed as AB or BA.
From the definition of subset, if , is still a subset of , and vice versa.
Three sets are defined as follows:
, and
We can see that CA and CB, as the elements and of are also elements
of and . However, AB, since some elements of are not elements of .
Based on what we have learned till now, some properties of sets are
1.
Every set is a subset of the universal set , since by definition, every element of a set belongs to .
2.
3.
4.
Two set are equal if and only if both are subsets of each other. In other words, if and only
ifAB and BA
5.
If every element of a set belongs to a set , and every element of set belongs to a set , then
clearly every element of set belongs to set . Thus, if AB and BC, then AC.
Proper Subset
If two sets have no elements in common, they are said to be disjoint. Two sets which are disjoint can never have
a superset-subset relationship, unless one is a null set.
Venn Diagrams
A venn diagram is a pictorial representation of sets, wherein they are shown as enclosed areas. Typically, the
universal set is represented by the area within a rectangle, and the other sets as circles placed within the
rectangle.
Set Operations
The three common operations on sets are the operations of union, intersection and difference.
Union
Fig 3: AB
The union of two sets, and , denoted by AB, is the set of all elements which belong to either or ; i.e.
AB={x:xA or xB}
The venn diagram depicting the union relationship is shown in Fig. 3.
Intersection
Fig 4: AB
The intersection of two sets, and , denoted by AB, is the set of all elements which belong to both and ;
i.e.
AB={x:xA and xB}
The venn diagram depicting the union relationship is shown in Fig. 4.
Three sets are defined as follows:
, and
AB={1,2,4,6,7,8,10,12}
BC={1,2,3,4,5,6,7,8,10,12}
AC={1,3,4,5,7,10}
AB={4,10}
BC=
AC={1,7}
Let
denote the set of students in a class. Let set and set denote the collection of boys and girls
We have,
BG=U, as each student in is either in set or set .
BG=, as there is no student belonging to both sets and .
Properties of unions and intersections
Some of the properties of the union and intersections of sets are as follows:
1.
(AB)A and (AB)B, since every element of AB belongs to , as well as . This is also clear from
the venn diagram of Fig. 4.
2.
A(AB) and B(AB), since every element of belongs to AB, and similarly every element
of belongs to AB. This is also clear from the venn diagram of Fig. 3.
Complement
The complement, or absolute complement, of a set , denoted by or , is the set of all elements which
belong to (the universal set) but does not belong to .A'={x:xU and xA}(see Fig. 5)
Fig 6: A~B
Difference
The difference of two sets, A and B, denoted by , or , is the set of all elements which belong
to but do not belong to , i.e.A~B={x:xA,xB}(see Fig. 6)