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Home / Algebra / Description of all Set, Some problems and solutions.

Description of all Set, Some problems and solutions.

Definition:A set may be viewed as any well defined collection of object. The objects are called the elements or members of the set.

A set will usually be denoted by a capital letter, such as,
A,B,X,Y,………………………
Whereas lower case letters a,b,c,x,y,z………. will usually be used to denote elements of sets.
Note that the elements are separated by comma enclosed in braces {}.
Example: A={a,e,i,o,u}

Types of sets: There are essentially two ways to specify a particular set,

(i) Tabular form of a set and
(ii) Set builder form of a set
Tabular form of a set:      If possible, is to list its element.

For example, A={a,e,i,o,u} ; means that A is the set whose elements are the letters a,e,i,o,u. Note that the elements are separated by comma enclosed in braces {}.
Set builder form of a set: Set builder form of a set is to state those properties which characterize the elements in the set, that is, the properties held the members of the set but not by nonmembers. Consider, for example, the expression
B={x:x is an even integer, x>0}
Which reads “ B is the set of x such that x is an even integer and x>0”
Equal set: Two set A and B are said to be equal if they have equal number of elements and it is written A=B.
Subset: If every elements of the set A belongs to set B, then A is said to be the Example: If A={1,3,5} and B= { 1,2,3,4,5,6} then A is a subset of B.
Universal set: All sets under investigation in any application of set theory are assumed o be contained in some large fixed set called the universal set or universe.
We will denote the universal set by U.
Union of sets: The set consisting of all the elements of two sets is called the union of two sets.
Union sets of A and B is denoted by A∪B and read A union B.
Example: Let A = {1,2,3} and B={4,5,6}
∵ A ∪B= {1,2,3,4,5,6}
Intersection of sets: The consisting of the common elements of two sets is called intersection of those two sets.
Intersection sets of A and B is denoted by A∩B and read A Intersection B.
Example: Let A= {2,3,4,5,6} and B= { 4,6,8}
∵ A∩B = { 4,6 }
Empty set: Let S={x:x is a positive integer, x2=3}
has no elements since no positive integer has the required property. This set with no elements is called the empty set or null set and is denoted by { } or ⊘.
Disjoint set: If two sets do not have any common element, then the sets are said to be disjoint.
If A and B are two disjoint set then A∩B=⊘.
Example: Let A= { 1,3,5} and B={2,4,6} then A∩B=⊘.
Complementary set: All sets under consideration at a particular time are subsets of a fixed universal set U. The absolute complement, or, simply complement of a set A, denoted by Ac, is the set of elements which belongs to U but which do not belongs to A ; that is
Ac={ x:x∈U, x∉A}.
Exercise: If U={1,2,3,4,5,6}, A= {1,2,3} and B={ 2,4,6}, then determine A′∩B′.
Solution: Given that,
U={1,2,3,4,5,6} ,
A= {1,2,3} and
B={ 2,4,6}.
Here, A′=U-A
= {1,2,3,4,5,6}-{1,2,3}
={4,5,6}
B′=U-B
={1,2,3,4,5,6}-{ 2,4,6}
={1,3,5}
Now , A′∩B′={4,5,6}∩{1,3,5}
= {5} Ans.
Exercise: If U={1,2,3,4,5,6,7,8,9}, A={1,2,5,6}, B= { 2,5,7} and C={ 1,3,5,7,9}, then determine (i) U∩( A ∪B)

(ii) (A∩B) ∪ (A∩C)
Solution: (i) Given that,
U={1,2,3,4,5,6,7,8,9},
A={1,2,5,6} ,
B= { 2,5,7} and
C={ 1,3,5,7,9}.
Here , A ∪B={1,2,5,6} ∪{ 2,5,7}
={1,2,5,6,7}
Now, U∩( A ∪B)= {1,2,3,4,5,6,7,8,9}∩{1,2,5,6,7}
= {1,2,5,6,7} Ans.
(ii) Given that,
U={1,2,3,4,5,6,7,8,9},
A={1,2,5,6} ,
B= { 2,5,7} and
C={ 1,3,5,7,9}.
Here, A∩B= {1,2,5,6}∩{ 2,5,7}
= {2,5}
A∩C= {1,2,5,6}∩{ 1,3,5,7,9}
= {1,5}

Now, (A∩B) ∪ (A∩C)= {2,5}∪{1,5}
= {1,2,5} Ans.

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