Let `X={1,2,3,4}`. Determine whether or not each relation is a function from X into X.
`h={(2,1),(3,4),(1,4),(4,4)}`

Answer

To determine whether the relation \( h = \{(2,1), (3,4), (1,4), (4,4)\} \) is a function from the set \( X = \{1, 2, 3, 4\} \) into itself, we need to check if every element in the domain \( X \) has a unique image in the codomain \( X \).

### Step-by-Step Solution:

1. **Identify the Domain and Codomain**:
- The domain is the set \( X = \{1, 2, 3, 4\} \).
- The codomain is also the set \( X = \{1, 2, 3, 4\} \).

2. **List the Elements of the Relation**:
- The relation \( h \) consists of the pairs: \( (2,1), (3,4), (1,4), (4,4) \).

3. **Check Each Element in the Domain**:
- We need to check if each element in the domain \( X \) (which are 1, 2, 3, and 4) appears as the first element (input) in any of the pairs in the relation \( h \).

- **For 1**:
- The pair \( (1,4) \) contains 1 as the first element.
- **For 2**:
- The pair \( (2,1) \) contains 2 as the first element.
- **For 3**:
- The pair \( (3,4) \) contains 3 as the first element.
- **For 4**:
- The pair \( (4,4) \) contains 4 as the first element.

4. **Check for Uniqueness**:
- Now, we need to ensure that each element in the domain maps to a unique element in the codomain:
- For 1: maps to 4 (from \( (1,4) \)).
- For 2: maps to 1 (from \( (2,1) \)).
- For 3: maps to 4 (from \( (3,4) \)).
- For 4: maps to 4 (from \( (4,4) \)).
- Notice that both 1 and 3 map to 4. This means that the element 4 in the codomain is not unique for the inputs 1 and 3.

5. **Conclusion**:
- Since the relation does not assign a unique output for every input (specifically, both 1 and 3 map to 4), the relation \( h \) is **not a function** from \( X \) into \( X \).

### Final Answer:
The relation \( h \) is not a function from \( X \) into \( X \).

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