Universal Set
The universal set is a set that contains all the elements related to a specific context. The universal set is denoted by U which is a superset of all sets with respect to a given context. For example, in the context set A represents the set of vowels, set B represents the set of consonants, set C represents the set of alphabets made up of straight lines only, etc, the universal set can be the set of all alphabets.
The universal set has important applications in probability where it is used to represent the sample space. Let us learn more about universal set along with its Venn diagram, its properties, and 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?
A universal set is a collection of all elements or members of all the related sets, known as its subsets. The set of all real numbers is the universal set in the context of sets of rational numbers, irrational numbers, integers, whole numbers, natural numbers, etc. In a particular context:
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.
Symbol of Universal Set
The universal set is represented by the symbol E or U. It consists of all the elements of its subsets, along with some extra 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 can be given by U = {1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12}.
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 (null 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 of sets and the universal set. We can understand the difference better by looking at their definitions.
| Universal Set | Union of Sets |
|---|---|
| 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) |
| Universal set may contain extra elements other than what its subsets have. | The union of two sets cannot have any extra elements other than the elements of the given two sets. |
Thus, the universal set is the union of all its subsets along with some extra elements in some cases. The following example can be used to understand this difference better. Consider three sets with elements 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 above two sets can be the set of all alphabets {a, b, c, d, …, z}
- 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.
Important Notes on Universal Set:
- If A is a subset of the universal set U, then all the elements in U that are not in A are called the complement of A (denoted by A').
- If A is a subset of the universal 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.
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Universal Set Examples
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Example 1: Given below is a Venn diagram representing the sets, A and B. Determine the elements of the universal set for its given subsets, A and B.
Solution:
We know that any universal set is represented by a rectangle and its subsets are represented by circles.
Here, the subset A = {3, 7, 9} and the subset B = {4, 8}. Clearly, A and B are disjoint sets because they have no common element. Also, the elements that are not contained in A and B are contained in the universal set.
Thus, the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}.
Answer: ∴ The universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}.
Example 3: Which of the following set(s) can be the universal set with the related sets as A = {1, 2, 3, 5}, B = {11, 13}, and C = {20, 21, 22}? Select all that apply.
(a) U = {1, 2, 3, …, 10}
(b) U = {1, 2, 3, …, 50}
(c) U = {1, 2, 3, …, 25}
(d) U = {1, 2, 3, …, 20}
Solution:
The given sets are A = {1, 2, 3, 5}, B = {11, 13}, and C = {20, 21, 22}.
The universal set in this context should contain all the elements of A, B, and C. Also, it may contain some different elements as well.
Using this definition, among the given options, only (b) U = {1, 2, 3, …, 50} (or) (c) U = {1, 2, 3, …, 25} can be the universal sets.
Answer: Options (b) and (c).
FAQs on Universal Set
What Is the Universal 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.
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.
What Is the Complement of the Universal 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 Do You Find Universal Sets?
We cannot define a unique universal set of some of its subsets are given. Because the universal set may contain some elements which none of its subsets have. Usually, a universal set is already defined in a context and we are not asked to determine it.
What Is the Difference Between a Universal Set and a Union of Seta?
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 of the given sets.
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 can be