what is set?and how many types of set?
Answers
Answer:
If a set contains only one element it is called to be a singleton set. Hence the set given by {1}, {0}, {a} are all consisting of only one element and therefore are singleton sets. Obviously, A, B contain a finite number of elements, i.e. 4 objects in A and 6 in B.
A set is a well-defined collection of distinct objects. The objects that make up a set (also known as the set's elements or members) can be anything: numbers, people, letters of the alphabet, other sets, and so on. ... Sets A and B are equal if and only if they have precisely the same elements.
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.
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