Q.Determine whether each of the following relations are reflexive, symmetric and transitive:

Properties of a Relation

A relation $R$ on a set $A$ pairs elements of $A$ with one another. Some relations behave in regular, predictable ways, and we name these behaviours properties. Three matter most for CBSE Class 12 — reflexive, symmetric, transitive (together they build an equivalence relation); a fourth, antisymmetric, is worth knowing for order relations.

Reflexive — everything relates to itself

$R$ is reflexive if $a\,R\,a$ for every $a\in A$. "Has the same age as" is reflexive; "is taller than" is not. If even one element misses its self-pair, reflexivity fails: on $\{1,2,3\}$, $\{(1,1),(2,2)\}$ is not reflexive because $(3,3)$ is absent.

Symmetric — the relation runs both ways

$R$ is symmetric if $a\,R\,b \implies b\,R\,a$. "Is married to" is symmetric; "is taller than" is not. Symmetry does not demand that every pair be related — only that any pair which appears also appears reversed. So $\{(1,2),(2,1),(3,3)\}$ is symmetric, but $\{(1,2),(2,1),(1,3)\}$ is not, since $(3,1)$ is missing.

Transitive — relations chain

$R$ is transitive if $a\,R\,b$ and $b\,R\,c$ together force $a\,R\,c$. "Is an ancestor of" is transitive; "is a friend of" is not. A single broken chain breaks the property: $\{(1,2),(2,3)\}$ is not transitive because $(1,3)$ is missing.

Antisymmetric — two-way ties force equality

$R$ is antisymmetric if $a\,R\,b$ and $b\,R\,a$ together force $a=b$. The order relation $\le$ is antisymmetric: $a\le b$ and $b\le a$ give $a=b$. It does not ban self-pairs like $(1,1)$; it only rules out distinct elements related both ways.

The famous combinations are the equivalence relation (reflexive + symmetric + transitive), which sorts a set into disjoint classes of "equivalent" elements, and the partial order (reflexive + antisymmetric + transitive), which arranges elements in a hierarchy.

The reflexive, symmetric, transitive, and antisymmetric properties of a relation are introduced in CBSE Class 11 Relations and Functions and revisited more formally at the start of the CBSE Class 12 Mathematics syllabus. "Reflexive symmetric transitive relation examples" is one of the most searched topics in this unit, since correctly testing all three properties is a near-guaranteed board exam question.