Universal Set

The universal set is a collection that includes all the elements relevant to a particular context. It is often symbolized by “U” and acts as the superset of all other sets related to that context. For example, if we consider a context where one set represents vowels, another set represents consonants, and a third set represents alphabets made up of straight lines, the universal set could be the set of all alphabets.

The universal set is crucial in probability, where it represents the sample space. Let’s dive deeper into the universal set definition, its Venn diagram, properties, and examples.

What Is Universal Set?

The universal set is a collection of all elements that belong to any related set, which are called its subsets. For example, in the context of numbers, if we consider sets of rational numbers, irrational numbers, integers, whole numbers, and natural numbers, the universal set would be the set of all real numbers.

Also Check: Sets – Definition, Theory, Formula, and Properties | Complement of a Set

Universal Set Definition

The universal set definition explains that it is a set that contains all elements from any related sets. It is usually represented by the symbol “U” or “E”. For instance, in human population studies, the universal set might be the set of all people in the world, while the population of each country would be considered a subset of this universal set.

• The universal set can be either finite or infinite.
• An example of an infinite universal set is the set of all natural numbers.

Symbol of Universal Set

The universal set is typically represented by the symbol “U” or “E”. It includes all elements of its subsets, along with possibly some additional elements.

Universal Set Example

Consider an example with three sets: one set containing {2, 4, 6}, another set containing {1, 3, 7, 9, 11}, and a third set containing {4, 8, 11}. The universal set for these sets, called “U”, would include {1, 2, 3, 4, 6, 7, 8, 9, 11}. Here, all the elements from the three sets are part of “U” without any repetition, making the elements in the universal set unique. Therefore, these sets are considered subsets of the universal set.

Complement of Universal Set

For any subset of the universal set, its complement includes all elements of the universal set that are not in the subset. The universal set contains all elements of its related subsets, while the empty set, or null set, contains no elements. Thus, the complement of the universal set is the empty set, often represented by an empty curly bracket {} or the symbol “Φ”.

Venn Diagram of a Universal Set

Venn diagrams are commonly used to visually show the relationships between sets. They are graphical representations where the universal set is depicted as a rectangle, and its subsets are shown as circles within that rectangle. Below is an example of a Venn diagram for a universal set.

In this Venn diagram, the universal set includes the elements from 1 to 10. The universal set, denoted as U, contains these elements: {1, 2, 3, …, 10}. A subset of this universal set is represented by set A, which includes the even numbers: A = {2, 4, 6, 8, 10}.

Do Check: Word Problems on Sets

Difference Between Universal Set and Union of Sets

Students often confuse the universal set with the union of sets. To clarify, let’s compare their definitions:

Thus, the universal set can be seen as the union of all its subsets, possibly with some additional elements.

Universal Set Example:

Consider two sets:

• Set A = {a, b, c}
• Set B = {e, f, g}

The universal set U, in this case, could be the set of all alphabets: U = {a, b, c, d, …, z}.

The union of sets A and B would be:

• A ∪ B = {a, b, c, e, f, g}.

In this example, the universal set contains elements from A, B, and additional elements, while the union of A and B only contains elements from A and B.

Also Check: Types of Sets

Important Points on Universal Set

• If set A is a subset of the universal set U, then the elements in U that are not in A are called the complement of A, denoted as A’.
• The complement of a subset is also a subset of the universal set.
• The complement of a universal set is always an empty set.
• A set and its complement are disjoint sets.

In summary, understanding the universal set definition, what is a universal set, and examining universal set examples helps clarify the concept and its distinction from the union of sets.

FAQs on Universal Set

What Is a Universal Set in Math?

The universal set is the collection of all possible elements or members related to specific sets. It is often symbolized by E or U. For instance, if we consider the set of all prisms, the universal set would be the collection of all three-dimensional shapes. This is a basic universal set definition.

What Is a Universal Set and How Does It Relate to Subsets?

When every element in set A is also found in set B, we say that A is a subset of B. Subsets are essentially smaller sets formed from the universal set. It's important to remember that the universal set itself can be considered a subset of itself, though the elements in any subset are fewer than those in the universal set it comes from. This is a practical way to understand what is universal set.

How Is a Universal Set Represented in a Venn Diagram?

Venn diagrams are a common way to visually represent the relationship between sets. In these diagrams, the universal set is usually depicted as a rectangle, while its subsets are shown as circles within this rectangle. This helps in understanding the universal set example visually.

What Is the Complement of the Universal Set?

The complement of the universal set is essentially an empty set, also known as a null set. Since the universal set includes every possible element, its complement contains no elements. The empty set is often denoted by the symbol '{}'. This concept ties back to the universal set definition.

Can You Define a Universal Set from Its Subsets?

It's challenging to define a universal set based solely on its subsets, as the universal set might include elements not present in any of its subsets. Typically, the universal set is predefined within a given context, so there's no need to determine it separately. This clarifies what is universal set in different scenarios.

What Is the Difference Between a Universal Set and a Union of Sets?

The universal set is the complete collection of all elements in related sets, while the union of sets refers to a specific operation where the resultant set includes all elements from the given sets. Understanding this difference is key to grasping the universal set definition.

What Is the Universal Set of All Right Triangles?

All triangles have three sides and three angles, but they come in various types based on these characteristics. Therefore, the universal set for all right triangles could either be the set containing all polygons or just the set of all triangles. This serves as a universal set example within the context of geometry.