If A is the relation “less than” from A = {1,2, 3,4, 6, 7} to B = {1, 4, 5}, then `R^(-1)` equals

Options

A. {(4, 1), (5, 1), (2, 5), (1, 4), (1, 5)}
B. {(4, 1), (5, 1), (4, 2), (5, 2), (4, 3), (5, 3)}
C. {(4, 1), (5, 1), (4, 2), (5, 2), (4, 3), (5, 3), (5, 4)}
D. None of thes

Answer

To solve the problem, we need to find the inverse of the relation "less than" from set A to set B. Let's break it down step by step.

### Step 1: Define the sets
We have two sets:
- Set A = {1, 2, 3, 4, 6, 7}
- Set B = {1, 4, 5}

### Step 2: Define the relation "less than"
The relation "less than" (denoted as R) from set A to set B includes all pairs (x, y) such that x ∈ A and y ∈ B, and x < y.

### Step 3: Identify the pairs in the relation R
Now, we will find all pairs (x, y) where x is from set A and y is from set B, satisfying the condition x < y:

- For x = 1: y can be 4, 5 → Pairs: (1, 4), (1, 5)
- For x = 2: y can be 4, 5 → Pairs: (2, 4), (2, 5)
- For x = 3: y can be 4, 5 → Pairs: (3, 4), (3, 5)
- For x = 4: y cannot be any element in B since there is no y such that 4 < y.
- For x = 6: y cannot be any element in B since there is no y such that 6 < y.
- For x = 7: y cannot be any element in B since there is no y such that 7 < y.

So, the relation R is:
R = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}

### Step 4: Find the inverse relation R^(-1)
The inverse relation R^(-1) consists of pairs (y, x) where (x, y) is in R. Thus, we will switch the elements in each pair:

From R:
- (1, 4) becomes (4, 1)
- (1, 5) becomes (5, 1)
- (2, 4) becomes (4, 2)
- (2, 5) becomes (5, 2)
- (3, 4) becomes (4, 3)
- (3, 5) becomes (5, 3)

So, the inverse relation R^(-1) is:
R^(-1) = {(4, 1), (5, 1), (4, 2), (5, 2), (4, 3), (5, 3)}

### Final Answer
Thus, the inverse relation R^(-1) is:
R^(-1) = {(4, 1), (5, 1), (4, 2), (5, 2), (4, 3), (5, 3)}

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