Universal set is a larger set formed fromtwo or more sets, which are also considered as the subsets of the universal set. Theuniversal set isdenoted by U and is a set containing all the elements of all its subsets, and the elements are not repeated. Consider two sets, A = {x,y,z} and B = {1,2,3,x,y}, then the universal set associated with these two sets is U = {1,2,3,x,y,z}.
The elements in the universal set are not repeated, thus they are unique. In this article, let's learn about the universal set, its definition, representation with solved examples.
| 1. | What Is Universal Set? |
| 2. | Venn Diagram of Universal Set |
| 3. | Difference Between Universal Set and Union of Set |
| 4. | Universal Set Examples |
| 5. | FAQs on Universal Set |
What Is Universal Set?
Universal set is a collection of all elements or members of all the related sets, known as its subsets. All the stars in a milky way galaxy is a good example ofa universal set if we consider all the stars in the milky way galaxy. When we study numbers in mathematics, we are interested in the set of natural numbers. This basic set is considered auniversal set and its subsets are even numbers, prime numbers, etc.
Universal Set Definition
The universal set is the set of all elements or members of all related sets. It is usually denoted by the symbol E or U. For example, in human population studies, the universal set is the set of all the people in the world. The set of all people in each country can be considered as a subset of this universal set.
- A universal set can be either a finite or infinite set.
- The set of natural numbers is a typical example of an infinite universal set. Set of natural numbers: {1,2,3,...}. Here, the ellipsis mark (...) denotes that the set goes on without an end.
Symbol of Universal Set
The universal set is usually represented by the symbol E or U. It consists of all the elements of its subsets, including its own elements.
Example of Universal Set
Let's consider an example with three sets, A, B, and C. Here, A = {2, 4, 6}, B = {1, 3, 7, 9, 11}, and C = {4, 8, 11}. We need to find the universal set for all three sets A, B, and C. All the elements of the given sets are contained in the universal set. Thus, the universal set U of A, B, and C is given by U = {1, 2, 3, 4, 6, 7, 8, 9, 11}
We can see that all the elements of the three sets are present in the universal set without any repetition. Thus, we can say that all the elements in the universal set are unique. The sets A, B, and C are contained in the universal set, then these sets are also called subsets of the Universal set.
- A ⊂ U (A is the subset of U)
- B ⊂ U (B is the subset of U)
- C ⊂ U (C is the subset of U)
Complement of Universal Set
For a subset A of the universal set (U), its complement is represented as A' which includes the elements of the universal set but not the elements of set A. The Universal set consists of a set of all elements of all its related subsets, whereas the empty set contains no elements of the subsets. Thus, the complement of the universal set is an empty set, denoted by ‘{}’ or the symbol 'Φ'.
Venn Diagram of Universal Set
Most of the time we use the Venn diagram to show the relationship between sets. Venn diagrams are the graphical representation of the sets. The universal set is represented by a rectangle and its subsets are represented by circles. Look at the Venn diagram shown below.
Here, we can see that the universal set U has these elements {1, 2, 3, ..., 10}, and the subset of this universal set is the set A = {2, 4, 6, 8, 10}.
Difference Between Universal Set and Union of Sets
Usually, students have confusion in differentiating between the union ofsets and the universal set. We can understand the difference better by looking at their definitions.
| Universal Set | Union of Set |
| The universal set is the set of all elements or members of all related sets. | The union of sets is one of the set operations between two sets where the resultant set contains all the elements belonging to both the initial sets. |
| A universal set can be denoted by the symbol U. | The union operation between sets can be denoted by the symbol ∪. Example: A ∪ B(A union B) |
The following example can be used to understand this difference better. Consider three sets with elements U = {3, 5}, set A = {a, b, c}, and set B = {e, f, g}. Let's find the universal set U and the union of sets A and B.
- The universal set of the 3 sets is given as follows: U = { a, b, c, e, f, g, 3, 5}
- The union between A and B is given as: A ∪ B = {a, b, c, e, f, g}
Thus, we can see that the universal set contains the elements from A, B, and U itself, whereas the union of A and B contains elements from only A and B.
Related Articles on Universal Set
Check out the following pages related to the universal set
- Finite and Infinite Sets
- Sets Formula
- Intersection of Sets
- Operation on Sets
Important Notes on Universal Set
Here is a list of a few important points related to universal set.
- If A is a subset of theuniversal set U, then all the elements in U that are not in A are called the complement of A.
- If A is a subset of theuniversal set U, then the complement of A is also a subset of U.
- The complement of a universal set is always an empty set.
- A set and its complement are the disjoint sets.
FAQs on Universal Set
What Is theUniversal Set in Math?
The universal set is the set of all elements or members of all the related sets. The universal set is usually denoted by the symbol E or U. For example, for the set of all kinds of prisms, the universal set is the set of all three-dimensional shapes.
What Are Universal Sets and Subsets?
If all the elements of set A are also the elements of another set B, then we can say that A is the subset of B. Then, the subsets can actually be created from any given universal set. We should also note that any universal set is actually a subset of itself. But the elements in a subset are less than the elements in the universal set from which the subset is created.
What Is the Complement of theUniversal Set?
The complement of the universal set can be considered as an empty set because when the universal set contains the set of all elements, then the empty set will contain no elements of the subsets. The null set is another term used for the empty set, and it is denoted by the symbol '{}'.
How To Represent Universal Set in a Venn Diagram?
Most of the time we use the Venn diagram to show the relationship between sets for more clarity. Venn diagrams are the graphical representation of the sets in which the universal set is represented by rectangles and its subsets are represented by circles.
How Do You Solve Universal Sets?
Let's consider an example with three sets, A, B, and C. Here, A = {2, 4, 6}, B = {1, 3, 7, 9, 11}, and C = {4, 8, 11}. We need to find the universal set for all three sets A, B, and C. All the elements of the given sets are contained in the universal set. Thus, the universal set U of A, B, and C is given by: U = {1, 2, 3, 4, 6, 7, 8, 9, 11}. We can see that all the elements of the three sets are present in the universal set without any repetition. Thus, we can say that all the elements in the universal set are unique.
What Is the Universal Set of All Right Triangles?
All triangles have three sides and three angles. There are different types of triangles based on their sides and angles. Thus, the universal set of all right triangles is the set containing all the polygons or a set of all polygons with three sides, that is all different types of triangles: equilateral Triangles, isosceles triangles, scalene triangles, acute-angled triangles, right-angled triangles, and obtuse-angled triangles.
What Is the Difference Between Universal Set and Union of Set?
The universal set is defined as a set containing all elements or members of all the related sets, known as its subsets, whereas the union of sets is one of the set operations between two sets where the resultant set contains all the elements which are common elements of both the initial sets.