Set notation refers to the various symbols used in the representation and operation of sets. It is a fundamental concept in mathematics, offering a structured and concise way to represent collections of objects, numbers, or elements. Set notation is used to define and operate on sets, including symbols like curly brackets, commas, colons, unions, intersections, set differences, and more.
Key elements of set notation:
- Curly Braces {}: Used to enclose the elements of sets.
- Elements: The individual objects or numbers within a set.
- Commas: Separate the elements within a set.
- Capital Letter: Typically used to name sets
Set Notation for Set Representation
This table represents a set using set notation, showing its elements and operations:
Set Notation for Set Representation | Symbol | Description |
|---|---|---|
Curly Brackets | {} | The curly brackets are used to represent a set. An example includes set A = {1, 2, 3}. |
Comma | , | The comma is used to separate the elements of the sets. |
Colon | : | It is used in the set-builder representation of a set. For example, S = {x: x is an even number} |
Element of | ∈ | It represents that an element belongs to the set. A = {1, 2} then 1∈ A. |
Not an Element of | ∉ | It represents that an element does not belong to a set. A = {2} then 1 ∉ A. |
U | It represents the universal set of a set | |
Φ | It represents the empty set. | |
Proper Subset | ⊂ | A set B is a proper subset of A if all elements of B are in A and A ≠ B. |
Subset | ⊆ | A set B is a subset of A if all elements of B are in A. |
Union | ∪ | The union of two sets includes all elements present in both sets. Example: A ∪ B = {x: x ∈ A or x ∈ B}. |
Intersection | ∩ | The intersection of two sets includes only the common elements between the two sets. Example: A ∩ B = {x: x ∈ A and x ∈ B}. |
Set Difference | - | The set difference includes the elements of the first set that are not present in the second set. Example: A - B = {x: x ∈ A and x ∉ B}. |
Symmetric Difference | Δ | The symmetric difference includes elements that are in either of the sets but not in both. Example: A Δ B = (A - B) ∪ (B - A). |
Complement of set A | A' | The complement of set A includes all elements of the universal set that are not in A. Example: A' = {x: x ∉ A}. |
Set Notation for Set Operations
Different set notations for set operations include union, intersection, subset, difference, symmetric difference, operation, and complement of sets.
Union
Union (U) is represented by ∪ set notation. Union is a binary operation 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 the first set that are not present in the 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 a set is represented by the ⊆ set notation. The set B is called a 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 a 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
(A’) of a set is represented by (Set)c in set notation. The complement of a set includes the elements of the universal set that are not present in the given set. It is mathematically represented as for a set A, Ac = {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}
Xc = {5, 6}
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:
- Clear and Precise: Makes mathematical ideas easy to understand and avoids confusion.
- Simplifies Complex Ideas: Breaks down complex concepts into simpler forms.
- Easy Operations: Helps with set operations like union and intersection, making problem-solving easier.
- Base for Advanced Topics: Key to learning advanced subjects like probability and calculus.
- Improves Logical Thinking: Builds problem-solving skills and logical reasoning.
- Used Everywhere: Applied in fields like computer science, economics, and engineering.
- Standardized: Provides a common way to discuss ideas, helping people collaborate.
- Makes Proofs Easier: Makes mathematical proofs clearer and more organized.
Solved Question on Set Notation
Example 1: Find the intersection for set P = {1, 3, 5} and Q = {2, 5, 8}.
Solution:
P = {1, 3, 5}
Q = {2, 5, 8}
P ∩ Q = {5}
Example 2: Find the union of set P = {5, 10} and Q = {12, 15, 18}.
Solution:
P = {5, 10}
Q = {12, 15, 18}
P ∪ Q = {5, 10, 12, 15, 18}
Example 3: Find the difference of set P = {1, 3, 5} and Q = {2, 5, 8}.
Solution:
P = {1, 3, 5}
Q = {2, 5, 8}
P - Q = {1, 3}
Example 4: Find the complement of set X = {a, b, d} and U = {a, b, c, d, e}.
Solution:
X = {a, b, d}
Xc = {c, e}
Example 5: Find whether P is a subset of Q or notA-B, where set P = {2, 4} and Q = {4, 5, 6}.
Solution:
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
Question 1: Find the union of two sets A ={6, 4} and B = {3, 10}.
Question 2:Find the intersection of two sets A = {5, 13} and B = {3, 13}.
Question 3: Find the set difference A - B wherea A = {12, 14, 16} and B = {8, 10, 13}.
Question 4: Find whether B is a subset of A or not, where A = {a, b, c} and B = {a, b}
Question 5:Find the complement of set A, given A = {2, 4} and U = {1, 2, 3, 4, 5}.
Answer:-
1. A∪B={3, 4, 6, 10}.
2. A∩B={13}.
3. A-B {12, 14, 16}.
4. B ⊆ A because all elements of B are in A.
5. A' = {1, 3, 5}.