Universal Set Overview
The universal set is created by combining two or more sets, which are also considered subsets of the universal set. It is typically denoted by the symbol U and contains all the elements of its subsets without any repetition. For instance, if we consider two sets A = {x,y,z} and B = {1,2,3,x,y}, the universal set associated with these two sets is U = {1,2,3,x,y,z}.
The elements within the universal set are unique and not repeated. In this article, we will delve into the definition and representation of the universal set, along with some solved examples.
What is a Universal Set?
The Universal Set is a set that comprises all the elements or members of all the related sets, known as its subsets. For instance, if we consider all the stars in the Milky Way galaxy, then it can be regarded as an example of a universal set. In mathematics, when we study numbers, the set of natural numbers is considered as the fundamental set or universal set, and its subsets are even numbers, prime numbers, etc.
Universal Set Definition
The set of all elements or members of related sets is called a universal set and is commonly represented by the symbols E or U. For instance, in studies related to the population of humans, the universal set can be defined as the set of all individuals in the world. Meanwhile, the set of individuals in each country can be considered a subset of this universal set.
- A universal set can either be finite or infinite.
- An example of an infinite universal set is the set of natural numbers, which is represented as {1,2, 3,...}. The ellipsis mark (...) signifies that the set continues without an end.
Consider that a Universal Set U consists of sets A, B, and C. These sets are referred to as Subsets of Universal Set U and are denoted as
- A ⊂U (A subset of U)
- B ⊂U (B subset of U)
- C ⊂U (C subset of U)
Universal Set Symbol
The symbol U or E represents the Universal Set, which comprises all the elements from its subsets, along with its own elements.
Universal Set Complement
In Set Theory, every set has a complement. The Empty Set is recognized as the complement of the Universal Set.
The Universal Set contains all the elements of its corresponding subsets. On the other hand, the Empty Set has no elements from its subsets. Thus, it is regarded as the complement of the Universal Set. The Empty Set is also known as the Null Set, and it is denoted by the symbol '{}' or 'Φ'.
Universal Set Venn Diagram
Venn Diagrams are used to show the relationship between sets in the form of pictorial representation. They are basically the graphical representation of sets.
- Universal Set is represented by arectangle in Venn Diagrams.
- The subsets are represented by circles or ovals.
Consider a universal set U along with its two subsets A and B.
- Universal Set U = {1, 2, 3, 4, 5, 6, 7, 8}.
- Set A = {2, 3, 4}
- Set B = {6, 7, 8}
Thus, the Venn Diagram of the Universal Set will be as follows:
Difference Between Universal Set & Union Set
The table below shows the difference between the Universal Set and the Union of Sets:
|
Universal Set |
Union of Sets |
|
|
Definition |
The set of all possible elements that can be used in a given context. |
The set of all elements that are in either of the given sets. |
|
Notation |
Usually denoted by a capital letter such as U or Ω. |
Denoted by the symbol ∪. |
|
Cardinality |
The cardinality of the universal set is typically infinite or unknown. |
The cardinality of the union of sets is the sum of the cardinalities of the individual sets, minus the cardinality of their intersection. |
|
Example |
The universal set of all possible outcomes in rolling a six-sided die is {1, 2, 3, 4, 5, 6}. |
The union of the sets {1, 2, 3} and {3, 4, 5} is {1, 2, 3, 4, 5}. |
|
Purpose |
The universal set serves as a reference for identifying the elements in a given context. |
The union of sets is used to identify all elements that are in either of the given sets, which can be useful for determining probabilities or finding commonalities between sets. |
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.
Universal Set: Important Points
- Universal Set is a type of set that includes all the elements of other sets, including its own elements.
- It is a larger set that contains elements of all the related sets, without any repetition.
- Universal Sets are dented by ‘U' or ‘E' in Set Theory.
- An Empty Set is the complement of a Universal Set.
- Union of Sets is different from Universal Set as it includes all elements belonging to either set or both.
- Sets are defined as well-organized collections of elements in Maths.
Universal Set: Sample Questions
What is a universal set, and how is it used in set theory?
Answer:A universal set is the set of all possible elements that can be used in each context. It is used as a reference for identifying the elements in a particular set.
What is the cardinality of a universal set?
Answer:The cardinality of a universal set is typically infinite or unknown, as it includes all possible elements.
If the universal set is {1, 2, 3, 4, 5}, and set A is {1, 3, 5}, what is the complement of set A?
Answer:The complement of set A is {2, 4}.
What is the difference between a universal set and an empty set?
Answer:A universal set includes all possible elements in each context, while an empty set contains no elements.
How does the intersection of sets relate to the universal set?
Answer:The intersection of sets is a subset of the universal set, as it includes only the elements that are common to both sets.