Let R be the real line. Consider the following subsets of the plane `RxxR` . `S""=""{(x ,""y)"":""y""=""x""+""1""a n d""0""<""x""<""2},""T""=""{(x ,""y)"":""x-y""` is an integer }. Which one of the following is true?

Options

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

Explanation

To determine whether the subsets \( S \) and \( T \) are equivalence relations, we need to check if they satisfy the properties of reflexivity, symmetry, and transitivity.

### Step 1: Analyze the set \( S \)

The set \( S \) is defined as:
\[ S = \{ (x, y) : y = x + 1 \text{ and } 0 < x < 2 \} \]

**Check Reflexivity:**
For a relation to be reflexive, every element \( (x, x) \) must belong to the relation.

- If we take any \( x \) in the interval \( (0, 2) \), we check if \( (x, x) \) belongs to \( S \):
\[ y = x + 1 \implies x \neq x + 1 \]
This is a contradiction. Therefore, \( (x, x) \notin S \) for any \( x \).

Thus, \( S \) is **not reflexive**.

### Step 2: Analyze the set \( T \)

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

**Check Reflexivity:**
For reflexivity, we need \( (x, x) \) to belong to \( T \):
\[ x - x = 0 \text{ which is an integer} \]
Thus, \( (x, x) \in T \) for all \( x \). Therefore, \( T \) is **reflexive**.

**Check Symmetry:**
For symmetry, if \( (x, y) \in T \), then we must show \( (y, x) \in T \):
- If \( x - y \) is an integer, then:
\[ y - x = -(x - y) \text{ which is also an integer} \]
Thus, \( (y, x) \in T \). Therefore, \( T \) is **symmetric**.

**Check Transitivity:**
For transitivity, if \( (x, y) \in T \) and \( (y, z) \in T \), we need to show \( (x, z) \in T \):
- If \( x - y \) is an integer and \( y - z \) is an integer, then:
\[ (x - y) + (y - z) = x - z \text{ which is also an integer} \]
Thus, \( (x, z) \in T \). Therefore, \( T \) is **transitive**.

### Conclusion
Since \( S \) is not reflexive, it cannot be an equivalence relation. However, \( T \) is reflexive, symmetric, and transitive, so it is an equivalence relation.

The correct answer is that \( T \) is an equivalence relation on \( R \) while \( S \) is not.