Empty Relation and Universal Relation
Imagine a set of students: {Ravi, Sita, Gopal}. A relation on this set is a rule telling you which pairs of students are "connected" in some way. For example, "is friends with" might connect Ravi–Sita and Sita–Gopal, but not Ravi–Gopal.
Two extremes exist: either no pair is connected, or every possible pair is. These extremes have special names.
Empty Relation
The empty relation (written ∅ or ϕ) is the relation where no element is related to any other — not even to itself.
In the empty relation, for every pair (a,b), the statement "a is related to b" is false.
Example: Let A={1,2,3}. The empty relation contains no ordered pairs at all:
Rempty={}
Nothing is related to anything.
Intuition: a party where nobody shakes hands with anyone — not even with themselves.
Universal Relation
The universal relation (written A×A) is the relation where every element is related to every other element — including itself.
In the universal relation, for every pair (a,b), the statement "a is related to b" is true.
Example: Let A={1,2,3}. The universal relation contains all possible ordered pairs:
Runiversal={(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}
Intuition: a party where everyone shakes hands with everyone — including themselves.
Precise Definitions
Let A be any non-empty set.
Empty Relation: R=∅⊆A×A
Universal Relation: R=A×A
- The empty relation has zero ordered pairs — the smallest possible relation on A.
- The universal relation has ∣A∣2 ordered pairs — the largest possible relation on A.
Why Do They Matter?
These two relations are the boundary cases for every property you'll study — reflexivity, symmetry, transitivity.
- The empty relation is not reflexive (for a non-empty set), but is symmetric and transitive vacuously — there are no pairs to violate the rules.
- The universal relation is reflexive, symmetric, and transitive — an equivalence relation where everything is equivalent to everything. …