Types of Relations

1. Types of Relations

Recalling a Relation

For a non-empty set $A$, recall from Class XI that any subset $R \subseteq A \times A$ is called a relation in the set $A$ (a relation from $A$ to itself). If $(a,b) \in R$, we write $a\,R\,b$ and say "$a$ is related to $b$". Two extreme cases deserve names: the empty relation $R = \phi$ (no element is related to any element) and the universal relation $R = A \times A$ (every element is related to every element).

Three Defining Properties

In plain language: reflexivity says every element is related to itself; symmetry says the relation reads the same "forwards and backwards"; transitivity says related pairs can be chained together.

Worked Illustration — A Relation That Fails All Three

Let $A = \mathbb{N}$ and $R = \{(a,b) : a > b\}$ ("$a$ is strictly greater than $b$").

• Reflexive? $(a,a) \in R$ would need $a > a$, which is false for every $a$. Not reflexive.
• Symmetric? Take $(5,3) \in R$ since $5 > 3$; but $(3,5) \notin R$ since $3 \not> 5$. Not symmetric.
• Transitive? If $a > b$ and $b > c$, then $a > c$ (a standard order property). Transitive.

So $R$ is transitive only — a reminder that the three properties are logically independent of each other; a relation may hold any one, any two, all three, or none.

Checking a Relation Systematically

To classify any relation $R$ on a described set $A$:

1. Reflexive — verify $(a,a) \in R$ for every $a \in A$; a single missing diagonal pair kills reflexivity.
2. Symmetric — for every pair present, check its reverse is also present; a single one-way pair kills symmetry.
3. Transitive — hunt for a genuine chain $(a,b), (b,c) \in R$ and check whether $(a,c) \in R$; a single broken chain kills transitivity.