Step 1: Determine the elements of the relation R.

The relation R is defined by (x, y) ∈ R if and only if max{x, y} ∈ {3, 4}. This means that the maximum of x and y must be either 3 or 4.

We can list the pairs (x, y) that satisfy this condition:

• If max{x, y} = 3: (0, 3), (1, 3), (2, 3), (3, 3), (3, 0), (3, 1), (3, 2)
• If max{x, y} = 4: (0, 4), (1, 4), (2, 4), (3, 4), (4, 4), (4, 0), (4, 1), (4, 2), (4, 3)

Therefore, R = {(0, 3), (1, 3), (2, 3), (3, 3), (3, 0), (3, 1), (3, 2), (0, 4), (1, 4), (2, 4), (3, 4), (4, 4), (4, 0), (4, 1), (4, 2), (4, 3)}

Step 2: Count the number of elements in R.

By counting the pairs listed above, we find that there are 7 pairs where max{x, y} = 3 and 9 pairs where max{x, y} = 4. Therefore, the total number of elements in R is 7 + 9 = 16.

Thus, statement (S₁) is false, since it claims that the number of elements in R is 18.

Step 3: Check if R is symmetric.

A relation is symmetric if whenever (x, y) ∈ R, then (y, x) ∈ R. From the definition of R, max{x, y} = max{y, x}. Therefore, if (x, y) ∈ R, then max{x, y} ∈ {3, 4}, which implies max{y, x} ∈ {3, 4}, and thus (y, x) ∈ R. So, R is symmetric.

Step 4: Check if R is reflexive.

A relation is reflexive if (x, x) ∈ R for all x ∈ A. In this case, we need to check if max{x, x} ∈ {3, 4} for all x ∈ A.

For x = 0, max{0, 0} = 0, which is not in {3, 4}. Therefore, (0, 0) ∉ R, and R is not reflexive.

Step 5: Check if R is transitive.

A relation is transitive if whenever (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.

Consider (0, 3) ∈ R and (3, 4) ∈ R. Then max{0, 3} = 3 and max{3, 4} = 4, so both pairs are in R. However, max{0, 4} = 4, so (0, 4) ∈ R.

Consider (0,3) ∈ R and (3,0) ∈ R. We need to check if (0,0) ∈ R. Since max{0,0} = 0 which is not in {3,4}, (0,0) is not in R.

Consider (3,0) ∈ R and (0,4) ∈ R. We need to check if (3,4) ∈ R. Since max{3,4} = 4 which is in {3,4}, (3,4) is in R.

Consider (1,3) ∈ R and (3,2) ∈ R. We need to check if (1,2) ∈ R. Since max{1,2} = 2 which is not in {3,4}, (1,2) is not in R. Thus R is not transitive.

Therefore, R is not transitive.

Step 6: Evaluate the statements.

(S₁) is false because the number of elements in R is 16, not 18.

(S₂) is true because R is symmetric, not reflexive, and not transitive.