Let $$A = \{0, 1, 2, 3, 4, 5\}$$. Let R be a relation on A defined by $$(x, y) \in R$$ if and only if $$\max\{x, y\} \in \{3, 4\}$$. Then among the statements
$$(S_1)$$ : The number of elements in R is 18, and
$$(S_2)$$ : The relation R is symmetric but neither reflexive nor transitive

Set $$A = \{0,1,2,3,4,5\}$$ has six elements.
The relation $$R$$ is defined by$$(x,y) \in R \; \Longleftrightarrow \; \max\{x,y\} \in \{3,4\}.$$

Case 1: $$\max\{x,y\}=3$$
Both coordinates must lie in $$\{0,1,2,3\}$$ and at least one of them must be $$3$$.
Total ordered pairs with coordinates from $$\{0,1,2,3\}$$ are $$4 \times 4 = 16$$.
Pairs with both coordinates in $$\{0,1,2\}$$ (hence max < 3) are $$3 \times 3 = 9$$.
Hence the number of pairs with max $$3$$ is $$16-9 = 7$$.

The explicit pairs are
$$(3,0),(3,1),(3,2),(3,3),(0,3),(1,3),(2,3).$$

Case 2: $$\max\{x,y\}=4$$
Both coordinates must lie in $$\{0,1,2,3,4\}$$ and at least one of them must be $$4$$.
Total ordered pairs with coordinates from $$\{0,1,2,3,4\}$$ are $$5 \times 5 = 25$$.
Pairs with both coordinates in $$\{0,1,2,3\}$$ (max < 4) are $$4 \times 4 = 16$$.
Hence the number of pairs with max $$4$$ is $$25-16 = 9$$.

The explicit pairs are
$$(4,0),(4,1),(4,2),(4,3),(4,4),(0,4),(1,4),(2,4),(3,4).$$

Since no ordered pair can contain the element $$5$$ (that would make the maximum $$\ge 5$$), the total number of elements in $$R$$ is
$$7 + 9 = 16.$$
Statement $$S_1$$ claims $$18$$ elements, so $$S_1$$ is false.

Next, examine the properties of $$R$$.

Symmetric:
If $$(x,y) \in R$$, then $$\max\{x,y\} \in \{3,4\}$$. The same maximum equals $$\max\{y,x\}$$, so $$(y,x) \in R$$. Hence $$R$$ is symmetric.

Reflexive:
Reflexivity requires every $$(a,a)$$, $$a \in A$$, to be in $$R$$.
But $$(a,a) \in R \Longleftrightarrow a \in \{3,4\}$$.
Elements $$0,1,2,5$$ violate this, so $$R$$ is not reflexive.

Transitive:
To test transitivity, find a counter-example.
Take $$x=0,\,y=4,\,z=0$$.
$$(x,y)=(0,4) \in R \quad (\max=4),$$
$$(y,z)=(4,0) \in R \quad (\max=4),$$
but $$(x,z)=(0,0) \notin R \quad (\max=0).$$
Thus $$R$$ is not transitive.

Therefore $$R$$ is symmetric but neither reflexive nor transitive, so Statement $$S_2$$ is true.

Conclusion: $$S_1$$ is false and $$S_2$$ is true → Option C (only $$S_2$$ is true).