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Updated on 20th March, 2023 , 5 min read
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.
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.
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.
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
The symbol U or E represents the Universal Set, which comprises all the elements from its subsets, along with its own elements.
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'.
Venn Diagrams are used to show the relationship between sets in the form of pictorial representation. They are basically the graphical representation of sets.
Consider a universal set U along with its two subsets A and B.
Thus, the Venn Diagram of the Universal Set will be as follows:
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.
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.
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.
Answer:The cardinality of a universal set is typically infinite or unknown, as it includes all possible elements.
Answer:The complement of set A is {2, 4}.
Answer:A universal set includes all possible elements in each context, while an empty set contains no elements.
Answer:The intersection of sets is a subset of the universal set, as it includes only the elements that are common to both sets.
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By - Nikita Parmar 2024-09-06 10:59:22 , 6 min readA universal set is the set of all possible elements that can be used in a given context. It is often denoted by the symbol U.
A universal set serves as a reference for identifying the elements in a particular set. It can also be used to define complements, unions, and intersections of sets.
A universal set includes all possible elements in a given context, while an empty set contains no elements.
The cardinality of a universal set is typically infinite or unknown, as it includes all possible elements.
Every set is a subset of the universal set, as the universal set includes all possible elements.
A proper subset is a subset of the universal set that does not include all possible elements.
The complement of a set is the set of all elements in the universal set that are not in the original set.
Yes, a universal set can be finite, but it must include all possible elements in the given context.
A universal set is often denoted by the symbol U or Ω.
A universal set is often used to define the sample space in probability theory, which is the set of all possible outcomes of an experiment.