Let `A ={1,2,3,4}. ` Let ` f : A to A " and " g : A to A`
defined by `f : ={(1,4),(2,1),(3,3),(4,2)}` and
`g = {(1,3),(2,1),(3,2),(4,4)}`
Find (i) g o f (ii) f o g (iii) f o f .

Answer

To solve the problem, we need to find the compositions of the functions \( g \) and \( f \) defined on the set \( A = \{1, 2, 3, 4\} \).

The functions are defined as follows:
- \( f = \{(1, 4), (2, 1), (3, 3), (4, 2)\} \)
- \( g = \{(1, 3), (2, 1), (3, 2), (4, 4)\} \)

We will find:
1. \( g \circ f \)
2. \( f \circ g \)
3. \( f \circ f \)

### Step 1: Find \( g \circ f \)

The composition \( g \circ f \) means we apply \( f \) first and then \( g \).

- For \( x = 1 \):
- \( f(1) = 4 \)
- \( g(f(1)) = g(4) = 4 \)
- So, \( g \circ f(1) = 4 \)

- For \( x = 2 \):
- \( f(2) = 1 \)
- \( g(f(2)) = g(1) = 3 \)
- So, \( g \circ f(2) = 3 \)

- For \( x = 3 \):
- \( f(3) = 3 \)
- \( g(f(3)) = g(3) = 2 \)
- So, \( g \circ f(3) = 2 \)

- For \( x = 4 \):
- \( f(4) = 2 \)
- \( g(f(4)) = g(2) = 1 \)
- So, \( g \circ f(4) = 1 \)

Thus, we have:
\[ g \circ f = \{(1, 4), (2, 3), (3, 2), (4, 1)\} \]

### Step 2: Find \( f \circ g \)

Now we compute \( f \circ g \):

- For \( x = 1 \):
- \( g(1) = 3 \)
- \( f(g(1)) = f(3) = 3 \)
- So, \( f \circ g(1) = 3 \)

- For \( x = 2 \):
- \( g(2) = 1 \)
- \( f(g(2)) = f(1) = 4 \)
- So, \( f \circ g(2) = 4 \)

- For \( x = 3 \):
- \( g(3) = 2 \)
- \( f(g(3)) = f(2) = 1 \)
- So, \( f \circ g(3) = 1 \)

- For \( x = 4 \):
- \( g(4) = 4 \)
- \( f(g(4)) = f(4) = 2 \)
- So, \( f \circ g(4) = 2 \)

Thus, we have:
\[ f \circ g = \{(1, 3), (2, 4), (3, 1), (4, 2)\} \]

### Step 3: Find \( f \circ f \)

Now we compute \( f \circ f \):

- For \( x = 1 \):
- \( f(1) = 4 \)
- \( f(f(1)) = f(4) = 2 \)
- So, \( f \circ f(1) = 2 \)

- For \( x = 2 \):
- \( f(2) = 1 \)
- \( f(f(2)) = f(1) = 4 \)
- So, \( f \circ f(2) = 4 \)

- For \( x = 3 \):
- \( f(3) = 3 \)
- \( f(f(3)) = f(3) = 3 \)
- So, \( f \circ f(3) = 3 \)

- For \( x = 4 \):
- \( f(4) = 2 \)
- \( f(f(4)) = f(2) = 1 \)
- So, \( f \circ f(4) = 1 \)

Thus, we have:
\[ f \circ f = \{(1, 2), (2, 4), (3, 3), (4, 1)\} \]

### Summary of Results:
1. \( g \circ f = \{(1, 4), (2, 3), (3, 2), (4, 1)\} \)
2. \( f \circ g = \{(1, 3), (2, 4), (3, 1), (4, 2)\} \)
3. \( f \circ f = \{(1, 2), (2, 4), (3, 3), (4, 1)\} \)