If the mappings f and g are given by : `f={(1,2),(3,5),(4,1)} and g={(2,3),(5,1),(1,3)}`, write fog.

Answer

To find the composition of the functions \( f \) and \( g \), denoted as \( f \circ g \), we need to follow these steps:

### Step 1: Understand the Functions
We have two mappings:
- \( f = \{(1, 2), (3, 5), (4, 1)\} \)
- \( g = \{(2, 3), (5, 1), (1, 3)\} \)

### Step 2: Determine the Output of \( g \)
To find \( f \circ g \), we need to find the output of \( g \) for each input in the domain of \( g \) and then use that output as the input for \( f \).

#### 2.1: Calculate \( g(2) \)
From the mapping \( g \):
- \( g(2) = 3 \)

#### 2.2: Calculate \( g(5) \)
From the mapping \( g \):
- \( g(5) = 1 \)

#### 2.3: Calculate \( g(1) \)
From the mapping \( g \):
- \( g(1) = 3 \)

### Step 3: Find \( f(g(x)) \)
Now we will use the outputs from \( g \) to find the corresponding outputs from \( f \).

#### 3.1: Calculate \( f(g(2)) = f(3) \)
From the mapping \( f \):
- \( f(3) = 5 \)

#### 3.2: Calculate \( f(g(5)) = f(1) \)
From the mapping \( f \):
- \( f(1) = 2 \)

#### 3.3: Calculate \( f(g(1)) = f(3) \)
From the mapping \( f \):
- \( f(3) = 5 \)

### Step 4: Compile the Results
Now we can compile the results of \( f \circ g \):
- \( f(g(2)) = 5 \) gives us the pair \( (2, 5) \)
- \( f(g(5)) = 2 \) gives us the pair \( (5, 2) \)
- \( f(g(1)) = 5 \) gives us the pair \( (1, 5) \)

### Final Result
Thus, the composition \( f \circ g \) is:
\[ f \circ g = \{(2, 5), (5, 2), (1, 5)\} \]