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Types of a Set
MathematicsComputer EngineeringMCA
Sets can be classified into many types. Some of which are finite, infinite, subset, universal, proper, singleton set, etc.
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 X 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 } = { 8 }
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.