An empty relation is a specific type of relation where no elements from the given sets are connected according to the defined rule, such as the relation of "being sisters" within a set of students from a boys' school.
Understanding Empty Relations
In mathematics, particularly in set theory, a relation defines how elements from one set are linked to elements of another set. An empty relation, also known as a null relation, occurs when the condition or rule defining the relation cannot be satisfied by any pair of elements from the specified sets. It means that the relation, as a subset of the Cartesian product of the sets, is the empty set itself (∅).
Key Characteristics:
- No Pair Satisfies the Condition: For every possible pair (a, b) from the sets, the rule governing the relation
Ris always false. - Result is the Empty Set: The relation
Rwill contain no ordered pairs, making itR = ∅.
Practical Example: Sisters in a Boys' School
To illustrate an empty relation, consider a common real-world scenario:
Imagine a school environment and a specific group of students.
- Set A: Let's define Set A as the collection of all students in grade 8 of a boys' school. This means every student in Set A is male.
- Relation R: We define a relation
Rbetween studentsaandbwithin Set A as follows:
R = { (a, b) | a and b are sisters }
In this context, the relation R is an empty relation.
Why is it empty?
Because all students in a boys' school are male, it is fundamentally impossible for any two students (a and b) selected from this set to be sisters. The condition "a and b are sisters" can never be fulfilled, no matter which two students you choose from Set A. Therefore, there are no ordered pairs (a, b) that can be included in relation R, making R an empty set.
Here's a summary of this example:
| Component | Description |
|---|---|
| Set (A) | Students of grade 8 in a boys' school |
| Rule (R) | (a, b) where a and b are sisters |
| Outcome | Empty Relation (no pairs satisfy the rule) |
Other Illustrative Examples
To solidify understanding, consider these additional scenarios:
-
Mathematical Example:
- Let Set X = {1, 2, 3} and Set Y = {4, 5, 6}.
- Define a relation R = { (x, y) | x ∈ X, y ∈ Y, and x = y }.
- This relation
Ris empty because there are no common elements between Set X and Set Y. No number from {1, 2, 3} can be equal to a number from {4, 5, 6}.
-
Abstract Example:
- Consider a set of positive integers,
Z+ = {1, 2, 3, ...}. - Define a relation R = { (x, y) | x ∈ Z+, y ∈ Z+, and x is a negative number }.
- This relation
Rwould be empty because, by definition, all numbers inZ+are positive. Thus, no elementxcan ever satisfy the condition of being a negative number.
- Consider a set of positive integers,
Empty relations are fundamental in understanding the boundaries and possibilities within mathematical and logical relationships, highlighting conditions that are inherently impossible given the constraints of the sets involved.
