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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.
If max{x, y} = 3, then the possible pairs (x, y) are (0, 3), (1, 3), (2, 3), (3, 3), (3, 0), (3, 1), (3, 2).
If max{x, y} = 4, then the possible pairs (x, y) are (0, 4), (1, 4), (2, 4), (3, 4), (4, 4), (4, 0), (4, 1), (4, 2), (4, 3).
Therefore, the relation R consists of the following pairs:
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.
The number of elements in R is 7 + 9 = 16.
Therefore, statement (S1) is false since it claims that the number of elements in R is 18.
Step 3: Check if R is symmetric.
For R to be symmetric, if (x, y) ∈ R, then (y, x) ∈ R.
If max{x, y} = 3, then max{y, x} = 3. If max{x, y} = 4, then max{y, x} = 4.
Since max{x, y} = max{y, x}, if (x, y) ∈ R, then (y, x) ∈ R. Therefore, R is symmetric.
Step 4: Check if R is reflexive.
For R to be reflexive, (x, x) ∈ R for all x ∈ A.
Consider x = 0. max{0, 0} = 0, which is not in {3, 4}. Therefore, (0, 0) ∉ R.
Thus, R is not reflexive.
Step 5: Check if R is transitive.
For R to be transitive, if (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.
If R is transitive, then (0, 4) should be in R. max{0, 4} = 4, so (0, 4) ∈ R.
However, consider (0, 3) ∈ R and (3, 0) ∈ R. If R is transitive, then (0, 0) ∈ R. But max{0, 0} = 0, which is not in {3, 4}. Therefore, (0, 0) ∉ R.
Consider (3,0) ∈ R and (0,3) ∈ R. If R is transitive, then (3,3) ∈ R. max{3,3} = 3, so (3,3) ∈ R.
Consider (1,3) ∈ R and (3,2) ∈ R. If R is transitive, then (1,2) should be in R. max{1,2} = 2, which is not in {3,4}. Therefore, (1,2) ∉ R.
Thus, R is not transitive.
Step 6: Evaluate the statements.
(S1): The number of elements in R is 18. (False, the number of elements is 16)
(S2): The relation R is symmetric but neither reflexive nor transitive. (True)
Correct Answer: only (S2) is true<\/strong>