What is a set?

A set is a well-defined collection of distinct elements. Sets are typically denoted by uppercase letters, and their elements are enclosed within curly braces $\{\}$.

Elements and cardinality

Elements

Each individual object within a set is known as an element.

In set A, the elements are 1, 2, 3, 4, and 5 and in set B, the elements are "apple," "mango," and "orange."

Cardinality

The number of elements in a set is called its cardinality. The cardinality of a set is denoted by vertical bars.

For set $A$, the cardinality, denoted by, $|A|$ is 5 as the set contains five elements and the cardinality of set B, denoted by, $|B|$ is 3.

Order of set

Sets are unordered collections of elements. This means that the arrangement of elements within a set does not matter. For example, the sets $\{1, 2, 3\}$ and $\{3, 2, 1\}$ are identical since they contain the same elements, even though their order differs.

Representation of sets

Sets can indeed be represented in any one of the following three ways or forms:

1. Descriptive form

2. Set-builder form

3. Roster or tabular form

Descriptive form

In the descriptive form, a set is described in words and specified by providing a verbal description of its elements, making it easy to know which items are part of the set and which are not.

Set-builder form

Set-builder notation, also known as the rule form, is a notation used to describe a set by indicating the properties or conditions that its members must satisfy.

The symbol $|$ or $:$ stands for "such that."

Roster or tabular form

In the roster form, a set is represented by listing all its elements inside a pair of braces $\{\}$. This form is especially useful when dealing with a finite set or presenting elements as part of a table with additional information.

Note: You can learn more about

• Types of sets

• Set operations

Properties

Commutative property

The commutative property applies to both union and intersection operations.

Associative property

The associative property applies to both union and intersection operations.

Distributive property

The distributive property applies to both union and intersection operations.

Identity property

The identity property applies to both union and intersection operations.

Complement property

The complement property applies to both union and intersection operations.

Idempotent property

The idempotent property applies to both union and intersection operations.

Conclusion

In conclusion, sets form the foundation of various mathematical concepts and operations. They are useful in solving problems across various fields, such as probability, logic, and computer science.