Let R be the real line. Consider the following subsets of the plane ` R xx R`. S={(x, y) y = x+1 and `0 lt x lt 2`} T = {(x, y):x-y is an integer). Which of the following is true ?

Options

A. Neither S nor T is an equivalence relation on R.
B. Both S and T are equivalence relations on R.
C. S is an equivalence relation but T is not
D. T is an equivalence relation but S is not

Answer

To solve the problem, we need to analyze the two sets \( S \) and \( T \) and determine whether they are equivalence relations. An equivalence relation must satisfy three properties: reflexivity, symmetry, and transitivity.

### Step 1: Analyze Set \( S \)

The set \( S \) is defined as:
\[ S = \{(x, y) : y = x + 1 \text{ and } 0 < x < 2\} \]
This means that for every \( x \) in the interval \( (0, 2) \), \( y \) is determined by the equation \( y = x + 1 \).

**Elements of \( S \):**
- If \( x = 0.1 \), then \( y = 1.1 \) → \( (0.1, 1.1) \)
- If \( x = 1 \), then \( y = 2 \) → \( (1, 2) \)
- If \( x = 1.5 \), then \( y = 2.5 \) → \( (1.5, 2.5) \)

**Reflexivity:**
To check reflexivity, we need to see if every element \( (a, a) \) is in \( S \).
- For \( (x, x) \) to be in \( S \), we would need \( x + 1 = x \), which is not possible. Therefore, \( S \) is **not reflexive**.

Since \( S \) is not reflexive, it cannot be an equivalence relation.

### Step 2: Analyze Set \( T \)

The set \( T \) is defined as:
\[ T = \{(x, y) : x - y \text{ is an integer}\} \]

**Reflexivity:**
For reflexivity, we need to check if \( (a, a) \) is in \( T \) for any real number \( a \).
- \( a - a = 0 \), and \( 0 \) is an integer. Thus, \( T \) is **reflexive**.

**Symmetry:**
To check symmetry, we need to see if \( (a, b) \in T \) implies \( (b, a) \in T \).
- If \( a - b \) is an integer, then \( b - a = -(a - b) \) is also an integer. Thus, \( T \) is **symmetric**.

**Transitivity:**
To check transitivity, we need to see if \( (a, b) \in T \) and \( (b, c) \in T \) implies \( (a, c) \in T \).
- If \( a - b \) is an integer and \( b - c \) is an integer, then \( (a - b) + (b - c) = a - c \) is also an integer. Thus, \( T \) is **transitive**.

Since \( T \) is reflexive, symmetric, and transitive, it is an equivalence relation.

### Conclusion

- Set \( S \) is **not an equivalence relation**.
- Set \( T \) **is an equivalence relation**.

Thus, the correct option is that \( S \) is not an equivalence relation, while \( T \) is.