1. What the Inverse of a Relation Means
The inverse of a relation reverses the direction of every ordered pair in the relation. If the original relation connects a → b, then the inverse connects b → a.
This gives a new relation formed by flipping each pair.
2. Formal Definition
If R is a relation from A to B, then its inverse R−1 is a relation from B to A.
2.1. Definition
R^{-1} = \{ (b,a) \mid (a,b) \in R \}
3. Understanding the Idea
The inverse simply swaps the positions of the elements in every ordered pair. The first element becomes the second, and the second becomes the first.
3.1. Visualization
(a,b) \in R \; \Rightarrow \; (b,a) \in R^{-1}
4. Example of Finding an Inverse Relation
Let:
R = \{(1,4),(2,5),(3,4)\}
To find R−1, reverse each pair:
4.1. Reversing Each Pair
- (1,4) → (4,1)
- (2,5) → (5,2)
- (3,4) → (4,3)
4.2. Final Inverse
R^{-1} = \{(4,1),(5,2),(4,3)\}
5. Inverse of Relations on the Same Set
If a relation is defined on a single set A (so R ⊆ A × A), then the inverse also lies in A × A.
In this case, the inverse shows how the relation looks when directions flip within the same set.
5.1. Example
R = \{(a,b),(c,a)\}
R^{-1} = \{(b,a),(a,c)\}
6. Properties of Inverses
Inverse relations behave in predictable ways:
6.1. Key Properties
- (R−1)−1 = R
- (R ∪ S)−1 = R−1 ∪ S−1
- (R ∩ S)−1 = R−1 ∩ S−1
- (S ∘ R)−1 = R−1 ∘ S−1
7. When a Relation Equals Its Inverse
If a relation looks the same after reversing all pairs, the relation is symmetric.
7.1. Connection to Symmetry
R \text{ is symmetric } \iff R = R^{-1}
8. Where Inverse Relations Are Useful
Inverse relations are used in reversing connections, undoing steps in mappings, understanding graphs, and studying functions (where only certain relations can have valid inverses).