Let's analyze the given relation $R = \{(1,1), (2,2), (3,3), (1,2), (2,3)\}$ in the set $S = \{1, 2, 3\}$, and prove whether it is reflexive, symmetric, and transitive.

1. Reflexive:

A relation $R$ on a set $S$ is reflexive if for every element $a \in S$, the pair $(a, a)$ is in $R$.

In this case, the set $S = \{1, 2, 3\}$. For $R$ to be reflexive, the following pairs must be present in $R$:

• $(1,1)$
• $(2,2)$
• $(3,3)$

We can see that all these pairs are in $R$. Therefore, the relation $R$ is reflexive.

2. Symmetric:

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

Let's check the pairs in $R$:

• $(1,1)$ is symmetric because $(1,1)$ is in $R$.
• $(2,2)$ is symmetric because $(2,2)$ is in $R$.
• $(3,3)$ is symmetric because $(3,3)$ is in $R$.
• $(1,2)$ is in $R$, but $(2,1)$ is not in $R$.
• $(2,3)$ is in $R$, but $(3,2)$ is not in $R$.

Since there are pairs for which the reverse pair is not present, the relation $R$ is not symmetric.

3. Transitive:

A relation $R$ on a set $S$ is transitive if for every three elements $a, b, c \in S$, whenever $(a, b) \in R$ and $(b, c) \in R$, we must have $(a, c) \in R$.

Let's check the pairs:

• $(1,2)$ and $(2,3)$ are in $R$, so by transitivity, $(1,3)$ must be in $R$. However, $(1,3)$ is not in $R$.

Since we found a counterexample, the relation $R$ is not transitive.

Conclusion:

• The relation $R$ is reflexive.
• The relation $R$ is not symmetric.
• The relation $R$ is not transitive.

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Would you like further details or clarifications? Here are some related questions for deeper understanding:

1. What is the definition of a symmetric relation with examples?
2. Can you provide a counterexample to show a non-reflexive relation?
3. How do you prove a relation is transitive or not with more complex sets?
4. How does the concept of equivalence relation relate to reflexive, symmetric, and transitive properties?
5. How does the composition of relations affect the transitivity property?

Tip: When checking for transitivity, look for chains of two pairs that share a common element, and verify if the corresponding third pair exists.