Find which of the following are the function on set `A={a,b,c} ?` Give reason.
(i) `R_(1)={(a,b),(b,a),(a,c)}`
(ii) `R_(2)={(b,b),(c,c),(a,b)}`

Answer

To determine whether the given relations \( R_1 \) and \( R_2 \) are functions on the set \( A = \{a, b, c\} \), we need to check if each element in the set \( A \) has a unique image in the relation.

### Step-by-Step Solution:

1. **Understanding the Definition of a Function:**
A relation \( R \) from set \( A \) to set \( A \) is a function if every element in \( A \) has exactly one image in \( A \). This means that for every \( x \in A \), there exists a unique \( y \in A \) such that \( (x, y) \in R \).

2. **Analyzing Relation \( R_1 = \{(a,b), (b,a), (a,c)\} \):**
- The pairs in \( R_1 \) are:
- \( a \) maps to \( b \)
- \( b \) maps to \( a \)
- \( a \) also maps to \( c \)
- Here, the element \( a \) has two images: \( b \) and \( c \). This violates the condition of having a unique image.
- Additionally, the element \( c \) does not appear in any pair, meaning it has no image.

**Conclusion for \( R_1 \):**
Since \( a \) does not have a unique image and \( c \) has no image at all, \( R_1 \) is **not a function**.

3. **Analyzing Relation \( R_2 = \{(b,b), (c,c), (a,b)\} \):**
- The pairs in \( R_2 \) are:
- \( b \) maps to \( b \)
- \( c \) maps to \( c \)
- \( a \) maps to \( b \)
- Here, every element in \( A \) has an image:
- \( a \) maps to \( b \)
- \( b \) maps to itself \( b \)
- \( c \) maps to itself \( c \)
- Each element has a unique image.

**Conclusion for \( R_2 \):**
Since every element in \( A \) has a unique image, \( R_2 \) is **a function**.

### Final Answer:
- \( R_1 \) is not a function.
- \( R_2 \) is a function.