Q.Give an example of a relation. Which is

The idea is that reflexivity, symmetry and transitivity are independent, so we can pick small relations on $A=\{1,2,3\}$ that satisfy exactly the combination asked for.

1. Symmetric, not reflexive, not transitive $R=\{(1,2),(2,1)\}$ — symmetric (each pair reversed); not reflexive ($(1,1)$ absent); not transitive ($(1,2),(2,1)$ need $(1,1)$, which is missing).
2. Transitive, not reflexive, not symmetric $R=\{(1,2)\}$ — transitive (no chain to violate it); not reflexive; not symmetric ($(2,1)$ absent).
3. Reflexive and symmetric, not transitive $R=\{(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)\}$ — reflexive and symmetric, but $(1,2),(2,3)$ need $(1,3)$, which is absent.
4. Reflexive and transitive, not symmetric $R=\{(1,1),(2,2),(3,3),(1,2)\}$ — reflexive; transitive (no chain forces a missing pair); not symmetric ($(2,1)$ absent). …

The trick for every part is the same: put in exactly the pairs that give the properties you want, and deliberately leave out a pair to kill the property you don't want. A single missing pair is enough to break reflexivity, symmetry or transitivity. Take $A=\{1,2,3\}$ throughout.

(i) Symmetric but neither reflexive nor transitive

• Symmetric: the only pair $(1,2)$ has its reverse $(2,1)$.
• Not reflexive: $(1,1)$ is missing.
• Not transitive: $(1,2)$ and $(2,1)$ would force $(1,1)$, which is absent.

(ii) Transitive but neither reflexive nor symmetric

• Transitive: there is no pair of the form $(1,2),(2,c)$, so the transitivity condition is never triggered — it holds vacuously.
• Not reflexive: $(1,1)$ is missing.
• Not symmetric: $(1,2)$ is present but $(2,1)$ is not.

(iii) Reflexive and symmetric but not transitive

• Reflexive: all of $(1,1),(2,2),(3,3)$ are present.
• Symmetric: $(1,2)\leftrightarrow(2,1)$ and $(2,3)\leftrightarrow(3,2)$.
• Not transitive: $(1,2)$ and $(2,3)$ would force $(1,3)$, which is missing.

(iv) Reflexive and transitive but not symmetric

• Reflexive: all three self-pairs are present.
• Transitive: the only non-self pair is $(1,2)$; the sole chain $(1,2),(2,2)$ needs $(1,2)$, already present. No pair is forced. …

Method: Constructing Relations with a Prescribed Combination of Properties

Use this when asked to build an example relation that has some properties but deliberately lacks others. Work on a small set such as $A=\{1,2,3\}$.

Steps

Step 1: Add pairs to create the properties you want

For symmetry, add both $(a,b)$ and $(b,a)$; for reflexivity, add all self-pairs $(1,1),(2,2),(3,3)$.

Step 2: Omit one pair to remove the property you don't want

Leave out a self-pair to break reflexivity, a reverse pair to break symmetry, or the "closing" pair of a chain to break transitivity.

Step 3: Use vacuous truth for the single-property cases …

Common Mistakes

Mistake 1: Building a symmetric and transitive relation that is accidentally reflexive

Why it's wrong: once $(a,b)$ and $(b,a)$ are present, transitivity forces $(a,a)$ and $(b,b)$; if every element appears, the relation becomes reflexive. Correct approach: exclude at least one element (e.g. $3$) entirely to keep it non-reflexive.

Mistake 2: Adding a pair that unintentionally completes a transitive chain …