Let's define a relation $R$ on the set $S = \{1, 2, 3, 4\}$ that is reflexive, not symmetric, and not transitive.

Reflexive Relation

A relation is reflexive if every element in the set is related to itself. This means that for all $x \in S$, the pair $(x, x)$ must be in the relation $R$.

So, for the set $S = \{1, 2, 3, 4\}$, the relation must at least contain the pairs: $R = \{(1, 1), (2, 2), (3, 3), (4, 4)\}$ This ensures that the relation is reflexive.

Not Symmetric

A relation is symmetric if for every pair $(a, b) \in R$, the pair $(b, a)$ must also be in $R$.

To make this relation not symmetric, we can add a pair $(1, 2)$ to the relation, but not $(2, 1)$. This breaks the symmetry.

So, we modify $R$ to: $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2)\}$ This relation is not symmetric because $(1, 2)$ is in $R$, but $(2, 1)$ is not.

Not Transitive

A relation is transitive if whenever $(a, b) \in R$ and $(b, c) \in R$, it implies that $(a, c) \in R$.

To make the relation not transitive, we can add $(1, 2)$ and $(2, 3)$ to the relation, but not $(1, 3)$. This will violate the transitivity condition because:

• We have $(1, 2)$ and $(2, 3)$ in the relation,
• But $(1, 3)$ is not in the relation, which means the relation is not transitive.

So, the final relation is: $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3)\}$

Conclusion

The relation $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3)\}$ is:

• Reflexive (because it contains $(x, x)$ for all $x \in S$),
• Not symmetric (because $(1, 2) \in R$ but $(2, 1) \notin R$),
• Not transitive (because $(1, 2) \in R$ and $(2, 3) \in R$, but $(1, 3) \notin R$).

Would you like further explanations or details on these properties? Here are some questions to dive deeper:

1. What is the difference between symmetric and asymmetric relations?
2. Can you provide an example of a relation that is symmetric and transitive?
3. How do you test for the reflexivity of a relation on a set?
4. What are the possible ways to break transitivity in a relation?
5. Can a relation be both symmetric and not reflexive?

Tip: When working with relations, it's useful to draw a diagram to visually represent which elements are related to each other. This can help you see symmetry, transitivity, and reflexivity more clearly.