Relations and Sets in Automata Theory

Quiz

In automata theory, we frequently use the set theory and its relations to map the models mathematically. In this chapter, we will understand them in detail. We will start from the basic relations of the domain and co-domain and learn different types of relations with notations.

At the end, we will see a quick reference table which will be useful to remember the most important relationship types.

Relations and Sets in Automata Theory

A relation in mathematics is a mapping between two sets of information, established by connecting elements of two or more non-empty sets.

A set of ordered pairs is a relation from set A into set B, where the ordered pairs are (1,1), (2,4), (3,9), and (4,16). The notation {1, 2, 3, 4} represents the domain and {1, 9, 16, 4} represents the range.

Types of Relations in Sets

Set us see some of the important relation types:

Empty Relation

An empty relation is a set where there is no relationship between any elements. For instance, R = {x, y} where |x – y| = 8 is an example of an empty relation. R = φ ⊂ A × A

Universal Relation

A universal relation is a set-wide relation where every element is related to each other. For example, R = {x, y} where |x – y| ≥ 0, where R = A × A. This type of relation is essential for understanding sets.

Identity Relation

An identity relation is a set where every element is related to itself, such as in a set A = {a, b, c}. It is represented by I = {(a, a), a ∈ A}.

Inverse Relation

Inverse relation refers to a set having elements that are inverse pairs of another set, such as R-1 = {(b, a): (a, b) ∈ R}, where R-1 = {(b, a): (a, b) ∈ R}.

Reflexive Relation

A reflexive relation is a set where every element maps to itself, as seen in the example of R = {(1, 1), (2, 2), (1, 2), (2, 1)}, where (a, a) ∈ R.

Symmetric Relation

A symmetric relation is a set where "a = b" is true, implying that "b = a" is also true. It is symmetric if (b, a) ∈ R is true when (a,b) ∈ R. An example is R = {(1, 2), (2, 1)} for a set A = {1, 2}. aRb ⇒ bRa, ∀ a, b ∈ A

Transitive Relation

The transitive relation (x, y, z) ∈ R implies (x, z) ∈ R, while aRb and bRc ⇒ aRc for all a, b, c ∈ A.

Equivalence Relation

An equivalence relation is a relation that is reflexive, symmetric, and transitive simultaneously.

For a quick reference, follow the table given below −

Type of Relation	Definition	Example
Relation	A relation R between sets S and T is a subset of S × T, and if an ordered pair (s, t) ∈ R, then an element in S is related to an element in T.	The relation R between elements s in S and t in T is defined as s is related to t if it is divided by s, and R = {(2, 12), (2, 16), (2, 18), (4, 12), (6, 12), (6, 18)}.
Reflexive	A relation R is said to be reflexive on a non-empty set R if every element of A is related to itself by that relation R.	Let A = {B, C, D}, where B, C, and D are brothers. The relation brotherhood is reflexive on the set A as {B, C}, {B, D}, and {C, D} are all sets of brothers.
Symmetric	A relation R is said to be symmetric, if for two elements 'a' and 'b' in X, if 'a' is related to 'b' then b is related to a.	Let A be a set of all students in a class. Let R be a relation called classmate. If a and b are two students belonging to the set, then a is a classmate of b and b is a classmate of a.
Transitive	Let R be a relation defined on a set A; then the relation R is said to be transitive if for a, b, c ∈ A and if aRb, bRc holds good then aRc also holds good.	Let A = {8, 6, 4}, where R is a relation called greater than. If 8 > 6 and 6 > 4 hold good, then 8 > 4 also holds good.
Equivalence relation	A relation R is called as an equivalence relation on 'A' if R is reflexive, symmetric, and transitive.	The set A = {a, b} is not an equivalence relation, as the (b, b) tuple is not present, but R = {(a, a), (a, b), (b, a), (b, b)} is.
Closure	A set is closed (under an operation) if and only if the operation on two elements of the set produces another element of the set. If an element outside the set is produced, then the operation is not closed.	The set S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10….} is a collection of integer numbers, and all elements of this set are included in it, indicating that integer numbers are closed under multiplication.

Conclusion

In this chapter, we explained relations in set theory and their types. Most important types are symmetric, reflexive, and transitive. The equivalence sets are combinations of them. We covered all of them with notations and explained the concepts examples for a better understanding.

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