Equality of Relations in Discrete Structures & Theory of Logic
In the realm of discrete structures and logic theory, the concept of equality in relations plays a significant role. Relations are fundamental in mathematics, especially in discrete mathematics, where they are used to describe connections or associations between elements of sets.
Understanding Relations
A relation R from set A to set B, denoted as R ⊆ A × B, is a set of ordered pairs where the first element belongs to set A, and the second element belongs to set B. In simpler terms, it represents a connection between elements of A and elements of B.
Equality in Relations
Equality in relations refers to when two relations are considered equal based on certain criteria. There are different criteria for equality in relations:
- Extensional Equality: Two relations R and S are extensionally equal if they contain the same set of ordered pairs. In other words, R = S if every element in R is also in S, and vice versa.
- Intensional Equality: Two relations R and S are intensionally equal if they are defined by the same rule or condition. Even if they contain different ordered pairs, they are considered equal if they represent the same relationship between elements.
Properties of Equal Relations
When discussing equality of relations, several properties come into play:
- Reflexivity: A relation R is reflexive if every element in the set is related to itself. Mathematically, ∀a ∈ A, (a, a) ∈ R.
- Symmetry: A relation R is symmetric if for every (a, b) ∈ R, (b, a) ∈ R. In simpler terms, if a is related to b, then b is related to a.
- Transitivity: A relation R is transitive if for every (a, b) ∈ R and (b, c) ∈ R, (a, c) ∈ R. This means that if a is related to b and b is related to c, then a is related to c.
Examples
Let's consider some examples to illustrate the concepts discussed above:
- Equality of Relations: Suppose we have two relations R = {(1, 2), (2, 3)} and S = {(2, 3), (1, 2)}. Even though the order of elements differs, R and S are extensionally equal because they contain the same set of ordered pairs.
- Reflexive Relation: If R = {(1, 1), (2, 2), (3, 3)} represents a relation on set A = {1, 2, 3}, then R is reflexive because every element is related to itself.
- Symmetric Relation: Consider the relation R = {(1, 2), (2, 1), (2, 2)}. R is symmetric because for every (a, b) ∈ R, (b, a) ∈ R.
- Transitive Relation: Let R = {(1, 2), (2, 3), (1, 3)} be a relation on set A = {1, 2, 3}. R is transitive because if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.