Set Notation

Set notation refers to the different symbols used in the representation and operation of sets. Set notation is a fundamental concept in mathematics, providing a structured and concise way to represent collections of objects, numbers, or elements. The set notation used to represent the elements of sets is curly brackets i.e., {}.

In this article, we will explore set notations for set representation and set operations. We will also cover the set notation table and solve some examples related to set notation.

What is Set Notation?

Set notations are the symbols used to represent the sets and operations on the sets. Different types of set notations are used in set theory. The set notation is used in every set like curly braces, commas, colons, unions, intersections, set differences and many more.

Set Notation for Set Representation

The different set notations for set representation include curly brackets, colon, belongs to, not belongs to, universal, and empty set.

Set Notation for Set Operations

Different set notation for set operations include union, intersection, subset, difference, symmetric difference and complement of sets.

Union

Union (U) is represented by ∪ set notation. Union is a binary operations on two sets that includes all the elements of both sets. It is mathematically represented as for two sets A and B, A ∪ B = {x: x∈A or x∈B}.

Example: Find the union of set X = {2, 3, 4} and Y = {4, 5, 6}.

Solution:

X = {2, 3, 4}

Y = {4, 5, 6}

X ∪ Y = {2, 3, 4, 5, 6}

Intersection

Intersection (∩) is represented by ∩ set notation. Intersection is a binary operation on two sets that includes the common elements of both sets. It is mathematically represented as for two sets A and B, A ∩ B = {x: x∈A and x∈B}.

Example: Find the intersection f set X = {2, 3, 4} and Y = {4, 5, 6}.

Solution:

X = {2, 3, 4}

Y = {4, 5, 6}

X ∩ Y = {4}

Difference

Difference ( \ ) is represented by \, - set notation. Difference is a binary operation on two sets that includes elements of first set that are not present in second set. It is mathematically represented as for two sets A and B, A - B = {x: x∈A and x∉B}.

Example: Find the difference of set X = {2, 3, 4} and Y = {4, 5, 6}.

Solution:

X = {2, 3, 4}

Y = {4, 5, 6}

X - Y = {2, 3}

Subset

The subset of set is represented by ⊆ set notation. The set B is called the subset of A if all the elements of set B are present in set A. It is mathematically represented as for two sets A and B, B ⊆ A = {x: x∈A ∀ x∈B}.

Example: Find whether X is subset of Y or not where set X = {2, 3, 4} and Y = {2, 3, 4, 5, 6}.

Solution:

X = {2, 3, 4}

Y = {2, 3, 4, 5, 6}

Since Y includes all the elements of X

Therefore, X ⊆ Y

Complement

Complement (A’) of set is represented by (Set)^c set notation. Complement of a set includes the elements of universal set that is not present in the given set. It is mathematically represented as for a set A A^c = {x: x∉A}.

Example: Find the complement of set X = {2, 3, 4} and U = {2, 3, 4, 5, 6}.

Solution:

X = {2, 3, 4}

X^c = {5, 6}

Set Notation for Set Operations Table

The table below represents different set notations used for set operations.

Set Notation Table

Table below represents the different set notations.

Benefits of Using Set Notation

Set notation offers several benefits that make it an essential tool in mathematics and related fields. Here are some of the key benefits:

1. Clarity and Precision: Set notation provides a clear and unambiguous way to represent and communicate mathematical ideas. It helps avoid confusion and ensures that statements and operations are well-defined.
2. Simplifies Complex Concepts: Using set notation, complex mathematical concepts and relationships can be expressed more simply and concisely. This makes it easier to understand and work with these concepts.
3. Facilitates Operations: Set notation allows for easy manipulation of sets through operations such as union, intersection, and difference. This simplifies problem-solving and logical reasoning in various areas of mathematics.
4. Foundation for Advanced Topics: Set notation is the basis for many advanced mathematical topics, including probability theory, calculus, and abstract algebra. Understanding set notation is crucial for progressing in these fields.
5. Enhances Logical Thinking: Working with sets and set notation develops logical thinking and reasoning skills. It encourages a structured approach to problem-solving and helps in constructing rigorous mathematical arguments.
6. Wide Applicability: Set notation is used in diverse areas such as computer science, economics, and engineering. Its universal applicability makes it a valuable tool across different disciplines.
7. Standardization: Set notation provides a standardized way to represent and discuss mathematical ideas, which is essential for collaboration and communication among mathematicians and scientists globally.
8. Efficiency in Proofs: Using set notation can streamline mathematical proofs by providing a compact and systematic way to express assumptions, intermediate steps, and conclusions.

Articles Related to Set Notation:

• Operation on Sets
• Set Theory Formulas
• De Morgan’s Law

Examples on Set Notation

1: Find the intersection for set P = {1, 3, 5} and Q = {2, 5, 8}.

P = {1, 3, 5}

Q = {2, 5, 8}

P ∩ Q = {5}

2: Find the union of set P = {5, 10} and Q = {12, 15, 18}.

P = {5, 10}

Q = {12, 15, 18}

P ∪ Q = {5, 10, 12, 15, 18}

3: Find the difference of set P = {1, 3, 5} and Q = {2, 5, 8}.

P = {1, 3, 5}

Q = {2, 5, 8}

P - Q = {1, 3}

4: Find the complement of set X = {a, b, d} and U = {a, b, c, d, e}.

X = {a, b, d}

X^c = {c, e}

5: Find whether P is subset of Q or not where set P = {2, 4} and Q = {4, 5, 6}.

P = {2, 4}

Q = {4, 5, 6}

Since Q does not includes all the elements of P (element 2)

Therefore, P is not a subset of Q.

Practice Questions on Set Notation

1. Find the union of two sets A ={6, 4} and B = {3, 10}.

2. Find the intersection of two sets A = {5, 13} and B = {3, 13}.

3. Find the set difference A - B where, A = {12, 14, 16} and B = {8, 10, 13}.

4. Find whether B is subset of A or not where A = {a, b, c} and B = {a, b}

5. Find the complement of set A given A = {2, 4} and U = {1, 2, 3, 4, 5}.