In Mathematics, sets are defined as the collection of objects whose elements are fixed and can not be changed. In other words, a set is well defined as the collection of data that does not carry from person to person. The elements can not be repeated in the set but can be written in any order. The set is represented by capital letters.
The empty set, finite set, equivalent set, subset, universal set, superset, and infinite set are some types of set. Each type of set has its own importance during calculations. Basically, in our day-to-day life, sets are used to represent bulk data and collection of data. So, here in this article, we are going to learn and discuss the universal set.
What is Set, What are Types of Sets, and Their Symbols?
A set is well defined as the collection of data that does not carry from person to person.
1. Empty Sets
The set, which has no elements, is also called a null set or void set. It is denoted by {}.
Below are the two examples of the empty set.
Example of empty set: Let set A = {a: a is the number of students studying in Class 6th and Class 7th}. As we all know, a student cannot learn in two classes, therefore set A is an empty set.
Another example of an empty set is set B = {a: 1 < a < 2, a is a natural number}, we know a natural number cannot be a decimal, therefore set B is a null set or empty set.
2. Singleton Sets
The set which has just one element is named a singleton set.
For example,Set A = { 8 } is a singleton set.
3. Finite and Infinite Sets
A set that has a finite number of elements is known as a finite set, whereas the set whose elements can’t be estimated, but has some figure or number, which is large to precise in a set, is known as infinite set.
For example, set A = {3,4,5,6,7} is a finite set, as it has a finite number of elements.
Set C = {number of cows in India} is an infinite set, there is an approximate number of cows in India, but the actual number of cows cannot be expressed, as the numbers could be very large and counting all cows is not possible.
4. Equal Sets
If every element of set A is also the elements of set B and if every element of set B is also the elements of set A, then sets A and B are called equal sets. It means set A and set B have equivalent elements and that we can denote it as:
A = B
For example, let A = {3,4,5,6} and B = {6,5,4,3}, then A = B
And if A = {set of even numbers} and B = { set of natural numbers} then A ≠ B, because natural numbers consist of all the positive integers starting from 1, 2, 3, 4, 5 to infinity, but even numbers start with 2, 4, 6, 8, and so on.
5. Subsets
A set S is said to be a subset of set T if the elements of set S belong to set T, or you can say each element of set S is present in set T. Subset of a set is denoted by the symbol (⊂) and written as S ⊂ T.
We can also write the subset notation as:
S ⊂ T if p ∊ S ⇒ p ∊ T
According to the equation given above, “S is a subset of T only if ‘p’ is an element of S as well as an element of T.” Each set is a subset of its own set, and a void set or empty set is a subset of all sets.
6. Power Sets
The set of all subsets is known as power sets. We know the empty set is a subset of all sets, and each set is a subset of itself. Taking an example of set X = {2,3}. From the above-given statements, we can write,
{} is a subset of {2,3}
{2} is a subset of {2,3}
{3} is a subset of {2,3}
{2,3} is also a subset of {2,3}
Therefore, power set of X = {2,3},
P(X) = {{},{2},{3},{2,3}}
7. Universal Sets
A set that contains all the elements of other sets is called a universal set. Generally, it is represented as ‘U.’
For example, set A = {1,2,3}, set B = {3,4,5,6}, and C = {5,6,7,8,9}.
Then, we will write the universal set as, U = {1,2,3,4,5,6,7,8,9,}.
Note: According to the definition of the universal set, we can say that all the sets are subsets of the universal set.
Therefore,
A ⊂ U
B ⊂ U
And C ⊂ U
8. Disjoint Sets
If two sets X and Y do not have any common elements, and their intersection results in zero(0), then set X and Y are called disjoint sets. It can be represented as;, X ∩ Y = 0.
Union, Intersection, Difference, and Complement of Sets
1. Union of Sets
The union of two sets consists of all their elements. It is denoted by (⋃).
For example, set A = {2,3,7} and set B = { 4,5,8}.
Then the union of set A and set B will be:
A ⋃ B = {2,3,7,4,5,8}
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2. Intersection of Sets
The set of all elements, which are common to all the given sets, gives an intersection of sets. It is denoted by ⋂.
For example, set A = {2,3,7} and set B = {2,4,9}.
So, A ⋂ B = {2}
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3. Difference of Sets
The difference between set S and set T is such that it has only those elements which are in the set S and not in the set T. S – T = {p : p ∊ S and p ∉ T}
Similarly, T – S = {p: p ∊ T and p ∉ S}.
4. Complement of a Set
Let U be the universal set and let A ⊂ U. Then, the complement of A, denoted by A’ or (U – A), is defined as
A’ = {x U : x A}
X A’ x A
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Every set has a complement of sets. Also, for a universal set, the empty set is known as the complement of the universal set. The empty set contains no elements of the subset and is also known as null set, which is denoted by {Ø} or {}.
Questions to be Solved
Question 1. If set A = {a, b, c, d} and B = {b, c, e, f} then, find A-B.
Answer: Let’s find the difference of the two sets,
A – B = {a, d} and B – A = {e, f}
Question 2. Let X = {David, Jhon, Misha} be the set of students of Class XI, who are in the school hockey team. Let Y = {Zoya, Rahul, Riya} be the set of students from Class XI who are in the school football team. Find X U Y and interpret the set.
Answer: (U union – combination of two sets)
Given X = {David, Jhon, Zoya}
Y = {Zoya, Rahul, Riya}
Common elements (Zoya) should be taken once
X U Y = {David, Jhon, Zoya, Rahul, Riya}.
This union set is equal to the set of students from Class XI who are present in the hockey team or the football team or both of the teams.