Let $$R = \{(3,3)(5,5),(9,9),(12,12),(5,12),(3,9),(3,12),(3,5)\}$$ be a relation on the set $$A = \{3, 5, 9, 12\}$$. Then, R is :

We are given a relation $$R = \{(3,3), (5,5), (9,9), (12,12), (5,12), (3,9), (3,12), (3,5)\}$$ on the set $$A = \{3, 5, 9, 12\}$$. We need to determine whether $$R$$ is reflexive, symmetric, and transitive.

First, recall the definitions:

• A relation is reflexive if every element in $$A$$ is related to itself, i.e., $$(a, a) \in R$$ for all $$a \in A$$.
• A relation is symmetric if whenever $$(a, b) \in R$$, then $$(b, a) \in R$$.
• A relation is transitive if whenever $$(a, b) \in R$$ and $$(b, c) \in R$$, then $$(a, c) \in R$$.

Now, let's check each property step by step.

Reflexivity: The set $$A = \{3, 5, 9, 12\}$$. Check if each element has a pair with itself in $$R$$:

• For 3: $$(3,3) \in R$$ (given).
• For 5: $$(5,5) \in R$$ (given).
• For 9: $$(9,9) \in R$$ (given).
• For 12: $$(12,12) \in R$$ (given).

Symmetry: Check for every pair $$(a, b) \in R$$, if $$(b, a) \in R$$. List all pairs:

• $$(3,3)$$: Reverse is $$(3,3)$$, which is in $$R$$.
• $$(5,5)$$: Reverse is $$(5,5)$$, which is in $$R$$.
• $$(9,9)$$: Reverse is $$(9,9)$$, which is in $$R$$.
• $$(12,12)$$: Reverse is $$(12,12)$$, which is in $$R$$.
• $$(5,12)$$: Reverse is $$(12,5)$$. Is $$(12,5) \in R$$? Looking at the given pairs, $$(12,5)$$ is not listed. So, $$(12,5) \notin R$$.
• $$(3,9)$$: Reverse is $$(9,3)$$. Is $$(9,3) \in R$$? Not listed, so $$(9,3) \notin R$$.
• $$(3,12)$$: Reverse is $$(12,3)$$. Is $$(12,3) \in R$$? Not listed, so $$(12,3) \notin R$$.
• $$(3,5)$$: Reverse is $$(5,3)$$. Is $$(5,3) \in R$$? Not listed, so $$(5,3) \notin R$$.

Transitivity: Check if whenever $$(a, b) \in R$$ and $$(b, c) \in R$$, then $$(a, c) \in R$$. We need to verify all possible chains:

• Consider $$(3,5)$$ and $$(5,12)$$: Both are in $$R$$. Then $$(3,12)$$ should be in $$R$$, and it is (given).
• Consider $$(3,3)$$ and $$(3,5)$$: Both in $$R$$. Then $$(3,5)$$ should be in $$R$$, and it is.
• Consider $$(3,3)$$ and $$(3,9)$$: Both in $$R$$. Then $$(3,9)$$ should be in $$R$$, and it is.
• Consider $$(3,3)$$ and $$(3,12)$$: Both in $$R$$. Then $$(3,12)$$ should be in $$R$$, and it is.
• Consider $$(3,5)$$ and $$(5,5)$$: Both in $$R$$. Then $$(3,5)$$ should be in $$R$$, and it is.
• Consider $$(5,5)$$ and $$(5,12)$$: Both in $$R$$. Then $$(5,12)$$ should be in $$R$$, and it is.
• Consider $$(5,12)$$ and $$(12,12)$$: Both in $$R$$. Then $$(5,12)$$ should be in $$R$$, and it is.
• Consider $$(3,9)$$ and $$(9,9)$$: Both in $$R$$. Then $$(3,9)$$ should be in $$R$$, and it is.
• Consider $$(3,12)$$ and $$(12,12)$$: Both in $$R$$. Then $$(3,12)$$ should be in $$R$$, and it is.

Now, check other possible chains:

• Is there a pair starting with 9? Only $$(9,9)$$, so no chain with another element.
• Is there a pair starting with 12? Only $$(12,12)$$, so no chain with another element.
• We have $$(3,9)$$ but no pair starting with 9 except $$(9,9)$$, so no issue.
• Similarly, $$(3,12)$$ and $$(12,12)$$ is covered.

No counterexample is found. Thus, $$R$$ is transitive.

In summary:

• Reflexive: Yes
• Symmetric: No
• Transitive: Yes

Now, compare with the options:

• A: reflexive, symmetric but not transitive → Incorrect, as $$R$$ is not symmetric.
• B: symmetric, transitive but not reflexive → Incorrect, as $$R$$ is reflexive and not symmetric.
• C: an equivalence relation → Incorrect, as equivalence requires all three properties, but symmetry fails.
• D: reflexive, transitive but not symmetric → Correct.