1. Given relation

We have $A=\{0,1,2,3,4,5\}$ and $(x,y)\in R \iff \max\{x,y\}\in \{3,4\}.$

So an ordered pair $(x,y)$ belongs to $R$ exactly when the larger of $x$ and $y$ is either $3$ or $4$.

2. Check statement (S1): number of elements in $R$

We count all ordered pairs $(x,y)$ such that $\max(x,y)=3$ or $\max(x,y)=4$.

Case 1: $\max(x,y)=3$

For this,

• both $x,y \le 3$
• and at least one of them must be $3$

Possible pairs: $(3,0),(3,1),(3,2),(3,3),(0,3),(1,3),(2,3)$

That is $7$ pairs.

(Equivalently: pairs from $\{0,1,2,3\}^2$ excluding those from $\{0,1,2\}^2$: $4^2-3^2=16-9=7$)

Case 2: $\max(x,y)=4$

For this,

• both $x,y \le 4$
• and at least one of them must be $4$

Possible pairs: $(4,0),(4,1),(4,2),(4,3),(4,4),(0,4),(1,4),(2,4),(3,4)$

That is $9$ pairs.

(Equivalently: $5^2-4^2=25-16=9$)

Total

$|R|=7+9=16$

So statement (S1): “The number of elements in $R$ is 18” is false.

3. Check statement (S2): symmetric, reflexive, transitive?

(i) Symmetric

Suppose $(x,y)\in R$. Then $\max(x,y)\in\{3,4\}.$ But $\max(x,y)=\max(y,x),$ so $\max(y,x)\in\{3,4\},$ which means $(y,x)\in R.$

Hence $R$ is symmetric.

(ii) Reflexive

A relation on $A$ is reflexive if $(a,a)\in R$ for every $a\in A$.

Now $(a,a)\in R \iff \max(a,a)=a\in\{3,4\}.$ So only $(3,3)$ and $(4,4)$ are in $R$.

But for example, $(0,0)\notin R, \quad (1,1)\notin R, \quad (2,2)\notin R, \quad (5,5)\notin R.$

Therefore $R$ is not reflexive.

(iii) Transitive

A relation is transitive if $(x,y)\in R \text{ and } (y,z)\in R \implies (x,z)\in R.$

We test by counterexample.

Take $(0,3)\in R$ because $\max(0,3)=3$, and $(3,5)\notin R$ so this won't help.

We need both first two pairs in $R$. Consider: $(0,3)\in R \quad \text{and} \quad (3,1)\in R$ since $\max(0,3)=3, \qquad \max(3,1)=3.$ Then transitivity would require $(0,1)\in R.$ But $\max(0,1)=1\notin\{3,4\},$ so $(0,1)\notin R.$

Hence $R$ is not transitive.

So statement (S2): “The relation $R$ is symmetric but neither reflexive nor transitive” is true.

4. Final evaluation of options

• (S1) is false
• (S2) is true

Therefore the correct option is: $\boxed{\text{C: only (S2) is true}}$

5. Comparison with stored correct answer

Stored correct answer: C

Our derived answer: C

So they agree.