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

Answer

To determine whether the relation \( g = \{(3,1), (4,2), (1,1)\} \) is a function from the set \( X = \{1, 2, 3, 4\} \) into itself, we need to follow these steps:

### Step 1: Understand the Definition of a Function
A relation is a function if every element in the domain (the first set) has exactly one image in the codomain (the second set). This means:
1. Every element from the domain must be paired with an element from the codomain.
2. No element in the domain can map to more than one element in the codomain.

### Step 2: Identify the Elements in the Domain
The elements in the domain for our relation \( g \) are the first elements of each ordered pair:
- From \( g \), the first elements are \( 3, 4, \) and \( 1 \).

### Step 3: Check for Completeness of the Domain
Now we check if every element from the set \( X \) is represented in the domain of the relation:
- The set \( X = \{1, 2, 3, 4\} \) contains the elements \( 1, 2, 3, \) and \( 4 \).
- The elements \( 3, 4, \) and \( 1 \) are present in the domain of \( g \), but the element \( 2 \) is missing.

### Step 4: Conclusion
Since the element \( 2 \) from the set \( X \) does not have a corresponding image in the relation \( g \), the relation does not satisfy the condition of a function.

Thus, we conclude that the relation \( g \) is **not a function** from \( X \) into \( X \).

### Final Answer
No, the relation \( g \) is not a function from \( X \) into \( X \) because the element \( 2 \) has no image in \( g \).

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