The relation R in the set $\{1, 2, 3\}$ given by $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 3)\}$ is:

The set is $\{1, 2, 3\}$ and the relation is $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 3)\}$.

A relation is reflexive if every element is related to itself. This requires $(1,1)$, $(2,2)$, and $(3,3)$ to be present.

$(1,1)$ is present

$(2,2)$ is present

$(3,3)$ is present

$R$ is reflexive.

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A relation is symmetric if whenever $(a,b)$ exists, $(b,a)$ must also exist.

$R$ contains $(1,2)$ but not $(2,1)$

$R$ contains $(1,3)$ but not $(3,1)$

$R$ is not symmetric.

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A relation is transitive if whenever $(a,b)$ and $(b,c)$ exist, $(a,c)$ must also exist.

$(1,2)$ and $(2,3)$ exist, need $(1,3)$ - present

$(1,2)$ and $(2,2)$ exist, need $(1,2)$ - present

$(1,1)$ and $(1,2)$ exist, need $(1,2)$ - present

$(1,1)$ and $(1,3)$ exist, need $(1,3)$ - present

$(2,2)$ and $(2,3)$ exist, need $(2,3)$ - present

$(2,3)$ and $(3,3)$ exist, need $(2,3)$ - present

All required pairs exist.

$R$ is transitive.

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Therefore, $R$ is reflexive and transitive.