A set is a collection of objects. The distinct entities form a group which have any kind of items, like, collection of alphabets, collection of numbers, days of the week, various colours, types of transportation, etc. This article will detail the definition, types of sets, and specifically equal and equivalent set symbols.
Sets
A set is defined by a collection of distinct items which are well-defined. Each item in a set is called an element of the set, and those items are enclosed within a set of curly brackets. For example, a set of prime numbers less than 10 is represented like: X = {1,2,3,5,7}.
This is the most basic example of set theory. Here, ‘X’ is how a set is represented; the letter must always be in capital.
Elements of a Set
The items that make up a set are called the elements of a set. They are separated by ‘,’ commas and put within a set of ‘{…}’ curly brackets. To represent if a certain element belongs to a set, ‘∈’ is used.
For example, in X = {1,2,3,5,7}, 5 ∈ X, meaning the element ‘5’ belongs to set ‘X’. If a certain item does not belong to the set, ‘∉’ is used, for example, 4 ∉ X, here the element ‘4’ is not the part of the set because 4 is not a prime number.
Cardinality in a Set
Cardinality, if explained in simple terms, is a total number of elements in a set. It is represented by n(X). For example, X = {1,2,3,5,7}, the cardinal number for this set would be n(X) = 5.
One of the most important conditions required for a set is that the items of the set should be co-related and must share a common type.
For example, if we say this set contains days of the week, another set contains months of the year, then a set is a collection of days and months, respectively.
Different types of Sets
This section will explore different types of sets:
- Finite Set
A finite set contains a definite number of well-defined items. For example, a set of days of the week would look like this:
D = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
- Infinite Set
An Infinite set contains elements that are not definite. For example, a set of even numbers would look like this:
E = {2, 4, 6, 8, 10, …}
E = { x|x ∈ N and x > 1}
- Subset
A subset is defined as a set that is a part of a bigger set. For example, set A is a subset of set B because each element of set A is in set B.
A = {1, 2, 3}
B = {1, 2, 3, 4, 5}
Here, elements of set A are a part of set B, so it can be represented as A ⊆ B.
- Universal Set
A universal set is a collection of objects which are university defined. All the sets are the proper subsets of the universal set because the universal set contains all possible elements concerning the shared common property.
‘U’ represents the universal set; for example, all even numbers belong to a universal set; likewise, the set of whole numbers is a universal set.
- Empty Set or Null Set
An empty or null set goes by the literal meaning that it is a set containing no elements. The cardinal number of an empty set is 0 and is called a finite set because 0 is a finite value. It is represented by ‘∅’.
For example, B = {x|x ∈ N, 2 < x < 3} = ∅
- Unit Set
A unit set is a kind of set which has a single element in a set. It is also known as a singleton set. A unit set is denoted by { s }, where ‘s’ represents the single element of a set.
For example, S = {x|x ∈ N, 2 < x < 4} = { 3 }
- Overlapping Set
Overlapping sets are sets that have at least one common element between two or more sets.
For example,
- n(X ∪ Y) = n(X) – n(X ∩ Y) n(Y)
- n(X ∪ Y) = n(X – Y) n(X ∩ Y) n(Y – X)
- n(X) = n(X – Y) n(X ∩ Y)
- n(Y) = n(X ∩ Y) n(Y – X)
- Disjoint Set
Disjoint sets are the opposite of what overlapping sets are. Disjoint sets do not contain any common items between them. Set X and Y are said to be disjoint sets when the elements of set X do not share any common element with set Y.
For example,
- n(X ∩ Y) = ∅
- n(X ∪ Y) = n(X) n(Y)
Equal and Equivalent Sets
The two sets having the same elements are known as equal sets. The order of the elements in both sets does not matter; the cardinality and the elements matter.
For example, set A and B are equal sets when,
A = {1,2,3,5,7}, B = {2,3,1,7,5}, here, each element is exactly similar to one another.
An equivalent set is a set that has a cardinal number similar to that of another set.
For example, set X and Y are equivalent sets when,
X = {A, B, C}, Y = {Monday, Tuesday, Wednesday}
Here, even though the elements in both sets are completely different from each other, the cardinality of both sets are equal. These kinds of sets are known as equivalent sets. n(X) = n(Y) = 3.
Sets Symbols
Here is the list of commonly used symbols which represent different kinds of sets:
| ‘U’ | Universal set |
| n(A) | The cardinality of set A |
| a ∈ X | ‘a’ is an element of set X |
| b ∉ Y | ‘b’ is not an element of set Y |
| ∅ | Empty set |
| A ∩ Y | Set A intersection set Y |
| C ⊆ D | Set C is a subset of set D |
Conclusion
A few things to notice while learning about sets is knowing how to identify the common and shared property that is essential to form a set. Two sets can only be equal sets when each element matches the elements of another set. Order is not necessary. While the equivalent sets have the same cardinality, and elements do not need to match.
