In the set X = {a, b, c, d}, which of the following functions in X?

Options

A. `R_(1) = {(b, a), (a, b), (c, d), (a, c)}`
B. `R_(2) = {(a, d), (d, c), (b, b), (c, c)}`
C. `R_(3) = {(a, b), (b, c), (c, d), (b, d)}`
D. `R_(4) = {(a, a), (b, b), (c, c), (a, d)}`

Explanation

To determine which of the given relations are functions from the set \( X = \{a, b, c, d\} \), we need to analyze each relation based on the definition of a function. A relation is a function if every element in the domain (in this case, the set \( X \)) is related to exactly one element in the codomain.

### Step-by-Step Solution:

1. **Identify the Relations**: We have four relations to analyze: \( R_1, R_2, R_3, R_4 \).

2. **Check Relation \( R_1 \)**:
- Elements: \( \{b, a, c, a\} \) (with \( a \) repeated)
- Mappings:
- \( b \to a \)
- \( a \to b \)
- \( a \to c \) (duplicate mapping for \( a \))
- \( c \to d \)
- Analysis:
- The element \( a \) is mapped to both \( b \) and \( c \).
- Since \( a \) has two images, \( R_1 \) is **not a function**.

3. **Check Relation \( R_2 \)**:
- Elements: \( \{a, d, b, c\} \)
- Mappings:
- \( a \to d \)
- \( d \to c \)
- \( d \to b \)
- \( c \to c \)
- Analysis:
- Each element \( a, d, b, c \) has a unique image.
- Therefore, \( R_2 \) is a **function**.

4. **Check Relation \( R_3 \)**:
- Elements: \( \{a, b, c\} \)
- Mappings:
- \( a \to b \)
- \( b \to c \)
- \( c \to d \)
- \( b \to d \) (duplicate mapping for \( b \))
- Analysis:
- The element \( b \) is mapped to both \( c \) and \( d \).
- Since \( b \) has two images, \( R_3 \) is **not a function**.

5. **Check Relation \( R_4 \)**:
- Elements: \( \{a, b, c, a\} \)
- Mappings:
- \( a \to a \)
- \( b \to b \)
- \( c \to c \)
- \( a \to d \) (duplicate mapping for \( a \))
- Analysis:
- The element \( a \) is mapped to both \( a \) and \( d \).
- Since \( a \) has two images, \( R_4 \) is **not a function**.

### Conclusion:
The only relation that qualifies as a function is \( R_2 \).