If `A=(1,2,3,4)}`, then which of the following are functions from A to itself?

Options

A. `f_(1)={(x,y),:y=x+1}`
B. `f_(2)={(x,y),x+y gt 4}`
C. `f_(3)={(x,y): y lt x}`
D. `f_(4)={(x,y): x+y =5}`

Explanation

To determine which of the given relations are functions from the set \( A = \{1, 2, 3, 4\} \) to itself, we need to check each relation to see if it satisfies the definition of a function. A relation \( f \) is a function if every element \( x \) in the domain (in this case, set \( A \)) is associated with exactly one element \( y \) in the codomain (also set \( A \)).

### Step-by-Step Solution:

1. **Check the first function \( f_1 = \{(x, y) | y = x + 1\} \)**:
- For \( x = 1 \), \( y = 2 \) (valid).
- For \( x = 2 \), \( y = 3 \) (valid).
- For \( x = 3 \), \( y = 4 \) (valid).
- For \( x = 4 \), \( y = 5 \) (not valid, as 5 is not in \( A \)).
- **Conclusion**: \( f_1 \) is **not a function** from \( A \) to \( A \).

2. **Check the second function \( f_2 = \{(x, y) | x + y > 4\} \)**:
- For \( x = 1 \), \( y \) can be \( 4 \) (valid).
- For \( x = 2 \), \( y \) can be \( 3 \) or \( 4 \) (two possible values, not valid).
- For \( x = 3 \), \( y \) can be \( 2 \) (valid).
- For \( x = 4 \), \( y \) can be \( 1 \) (valid).
- **Conclusion**: \( f_2 \) is **not a function** from \( A \) to \( A \) because \( x = 2 \) gives two different \( y \) values.

3. **Check the third function \( f_3 = \{(x, y) | y < x\} \)**:
- For \( x = 1 \), \( y \) can be \( 0 \) (not valid).
- For \( x = 2 \), \( y \) can be \( 1 \) (valid).
- For \( x = 3 \), \( y \) can be \( 1 \) or \( 2 \) (two possible values, not valid).
- For \( x = 4 \), \( y \) can be \( 1, 2, \) or \( 3 \) (multiple values, not valid).
- **Conclusion**: \( f_3 \) is **not a function** from \( A \) to \( A \).

4. **Check the fourth function \( f_4 = \{(x, y) | x + y = 5\} \)**:
- For \( x = 1 \), \( y = 4 \) (valid).
- For \( x = 2 \), \( y = 3 \) (valid).
- For \( x = 3 \), \( y = 2 \) (valid).
- For \( x = 4 \), \( y = 1 \) (valid).
- Each \( x \) has a unique corresponding \( y \) in \( A \).
- **Conclusion**: \( f_4 \) is a **function** from \( A \) to \( A \).

### Final Answer:
The only function from \( A \) to itself is \( f_4 \).