If 'R' is relation 'less than' from :
Set A = {(1, 2, 3, 4, 5}` to Set B = {1, 4, 6},
write down the Cartesian Product corresponding to 'R'.
Also, find the inverse relation to 'R'.

Answer

To solve the problem step by step, we need to find the Cartesian Product corresponding to the relation 'less than' (denoted as 'R') from set A to set B, and then find the inverse relation of 'R'.

### Step 1: Define the Sets
We have:
- Set A = {1, 2, 3, 4, 5}
- Set B = {1, 4, 6}

### Step 2: Identify the Relation 'R'
The relation 'R' is defined as 'less than'. This means we will find pairs (x, y) such that x is an element of set A and y is an element of set B, and x < y.

### Step 3: Find the Pairs for Relation 'R'
We will check each element of set A against each element of set B to see if the relation holds:

- For x = 1:
- 1 < 1 (False)
- 1 < 4 (True) → (1, 4)
- 1 < 6 (True) → (1, 6)

- For x = 2:
- 2 < 1 (False)
- 2 < 4 (True) → (2, 4)
- 2 < 6 (True) → (2, 6)

- For x = 3:
- 3 < 1 (False)
- 3 < 4 (True) → (3, 4)
- 3 < 6 (True) → (3, 6)

- For x = 4:
- 4 < 1 (False)
- 4 < 4 (False)
- 4 < 6 (True) → (4, 6)

- For x = 5:
- 5 < 1 (False)
- 5 < 4 (False)
- 5 < 6 (True) → (5, 6)

### Step 4: Compile the Relation 'R'
From the above checks, we can compile the relation 'R':
- R = {(1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6), (4, 6), (5, 6)}

### Step 5: Write the Cartesian Product
The Cartesian Product corresponding to the relation 'R' is:
- R = {(1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6), (4, 6), (5, 6)}

### Step 6: Find the Inverse Relation 'R^(-1)'
The inverse relation 'R^(-1)' consists of the pairs (y, x) such that (x, y) is in R. We will switch the elements in each pair:

- From (1, 4) → (4, 1)
- From (1, 6) → (6, 1)
- From (2, 4) → (4, 2)
- From (2, 6) → (6, 2)
- From (3, 4) → (4, 3)
- From (3, 6) → (6, 3)
- From (4, 6) → (6, 4)
- From (5, 6) → (6, 5)

### Step 7: Compile the Inverse Relation 'R^(-1)'
The inverse relation 'R^(-1)' is:
- R^(-1) = {(4, 1), (6, 1), (4, 2), (6, 2), (4, 3), (6, 3), (6, 4), (6, 5)}

### Final Result
- The Cartesian Product corresponding to 'R' is:
R = {(1, 4), (1, 6), (2, 4), (2, 6), (3, 4), (3, 6), (4, 6), (5, 6)}

- The inverse relation 'R^(-1)' is:
R^(-1) = {(4, 1), (6, 1), (4, 2), (6, 2), (4, 3), (6, 3), (6, 4), (6, 5)}