1. Define Set.Write down the Properties of Set.
2.What are the methods commonly used to represent a set. Give example of each .
3. What are different types of sets ? Give example of each type.
Answers
Step-by-step explanation:
1.In Mathematics, a set is defined as a collection of well-defined objects. For example, the set of natural numbers between 1 and 10, the set of even numbers less than 20. Another set A union B denoted by A⋃B, is the set which contains all the elements of A and B. ...
2.Representation of Sets
The sets are represented in curly braces, {}. For example, {2,3,4} or {a,b,c} or {Bat, Ball, Wickets}. The elements in the sets are depicted in either the Statement form, Roster Form or Set Builder Form.
3.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.