Which of the following relation are functions? Give reasons. If it is a function, determine its domain and range.
`{(a,b),(b,c),(c,d),(d,e)}`

Answer

To determine if the given relation is a function and to find its domain and range, we will follow these steps:

### Step 1: Identify the given relation
The relation provided is:
\[ R = \{(a,b), (b,c), (c,d), (d,e)\} \]

### Step 2: Check if the relation is a function
A relation is a function if every element in the domain (the first component of each ordered pair) maps to exactly one element in the range (the second component of each ordered pair).

- In our relation:
- The first elements (domain) are: \( a, b, c, d \)
- The second elements (range) are: \( b, c, d, e \)

Now, let's check:
- \( a \) maps to \( b \)
- \( b \) maps to \( c \)
- \( c \) maps to \( d \)
- \( d \) maps to \( e \)

Since each element in the domain maps to a unique element in the range, we can conclude that \( R \) is indeed a function.

### Step 3: Determine the domain
The domain of the function consists of all the first elements of the ordered pairs:
\[ \text{Domain} = \{a, b, c, d\} \]

### Step 4: Determine the range
The range of the function consists of all the second elements of the ordered pairs:
\[ \text{Range} = \{b, c, d, e\} \]

### Final Conclusion
Thus, the relation \( R \) is a function, with:
- **Domain**: \( \{a, b, c, d\} \)
- **Range**: \( \{b, c, d, e\} \)

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