Types of Relations – From Intuition to Precision
Imagine you have a set of people in a room. A relation is simply a rule that tells you whether two people are connected in some way. "Is the brother of", "lives in the same city as", "is taller than" — each of these is a relation. The question is: what kind of connection is it?
Some relations are very special. They behave in predictable, almost perfect ways. These are the ones we study first.
The Intuition: Three Key Properties
Think about the relation "lives in the same city as" between people.
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Reflexive: Does every person live in the same city as themselves? Yes — obviously. A relation is reflexive if every element is related to itself.
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Symmetric: If A lives in the same city as B, does B live in the same city as A? Yes — it's mutual. A relation is symmetric if whenever A is related to B, B is also related to A.
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Transitive: If A lives in the same city as B, and B lives in the same city as C, does A live in the same city as C? Yes — it chains. A relation is transitive if whenever A is related to B and B is related to C, then A is related to C.
Now think about "is taller than". It is not reflexive (no one is taller than themselves), not symmetric (if A is taller than B, B is not taller than A), but it is transitive (if A > B and B > C, then A > C). Different relations satisfy different combinations.
The Precise Definitions
Let R be a relation on a set A (meaning R⊆A×A).
Reflexive: R is reflexive if ∀a∈A, (a,a)∈R.
Every element is related to itself.
Symmetric: R is symmetric if ∀a,b∈A, (a,b)∈R⟹(b,a)∈R.
If a is related to b, then b is related to a.
Transitive: R is transitive if ∀a,b,c∈A, (a,b)∈R∧(b,c)∈R⟹(a,c)∈R.
If a is related to b and b is related to c, then a is related to c.
The Four Types of Relations You Must Know
These three properties combine to define the most important types:
| Type | Reflexive? | Symmetric? | Transitive? | Example |
|---|
| Empty relation | No (unless A=∅) | Yes (vacuously) | Yes (vacuously) | R=∅ on A={1,2} |
| Universal relation | Yes | Yes | Yes | R=A×A |
| Identity relation | Yes | Yes | Yes | R={(a,a)∣a∈A} |
| Equivalence relation | Yes | Yes | Yes | "Same city", "same remainder mod 3" |
A common mistake: thinking "symmetric" means "if (a,b) is in R, then (b,a) must also be in R" — that's correct. But it does not require that (a,b) and (b,a) are different pairs. The identity relation is symmetric because (a,a) implies (a,a).
Equivalence Relations — The Most Important Type
An equivalence relation is one that is reflexive, symmetric, and transitive all at once. It captures the idea of "sameness" or "equivalence" in some sense.
An equivalence relation partitions the set into disjoint equivalence classes — groups of elements that are all related to each other. Every element belongs to exactly one class.
Example: On the set of integers, define a∼b if a−b is divisible by 3. This is an equivalence relation. The classes are:
- {…,−6,−3,0,3,6,…} (remainder 0) …