A relation $R$ on the set $A = \{1,2,3,4\}$ is reflexive if every element is related to itself, i.e., $(a, a) \in R$ for all $a \in A$.
It is symmetric if whenever $(a, b) \in R$, then $(b, a) \in R$.
It is not transitive if there exist elements $a, b, c \in A$ such that $(a, b) \in R$ and $(b, c) \in R$, but $(a, c) \notin R$.

Example:

Consider the relation
$R = \{(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (2,3), (3,2)\}$

Checking Properties:

1. Reflexive: Since $(1,1), (2,2), (3,3), (4,4)$ are in $R$, every element is related to itself. ✅
2. Symmetric: If $(1,2) \in R$, then $(2,1) \in R$; if $(2,3) \in R$, then $(3,2) \in R$. So, $R$ is symmetric. ✅
3. Not Transitive: $(1,2) \in R$ and $(2,3) \in R$, but $(1,3) \notin R$, which violates transitivity. ❌

Thus, this relation is reflexive, symmetric, but not transitive.

Would you like more examples or further explanations?

Related Questions:

1. Can a relation be symmetric and transitive but not reflexive?
2. What is an example of a relation that is transitive but not symmetric?
3. How do equivalence relations relate to these properties?
4. Can a relation be reflexive but neither symmetric nor transitive?
5. What happens if we add $(1,3)$ to the given relation?

Tip:

To check transitivity, look for cases where $(a,b)$ and $(b,c)$ exist but $(a,c)$ does not.