Let R be the relation from `A={1,2,3,4,5,6}` to `B={1,3,5}` which is defined by the open sentence ''x is less than y''.
(i) Find the solution set of R, that is write R as set of ordered pairs.
(ii) Plot R on a co-ordinate diagram of `AxxB`.
(iii) state the domaing, range and codomain of R.

Answer

To solve the problem step by step, we will follow the instructions provided in the question.

### Step 1: Find the solution set of R as a set of ordered pairs.

Given:
- Set A = {1, 2, 3, 4, 5, 6}
- Set B = {1, 3, 5}
- Relation R is defined by the condition "x is less than y", where x ∈ A and y ∈ B.

We will check each element in A against each element in B to find the ordered pairs (x, y) such that x < y.

1. For x = 1:
- 1 < 1 (False)
- 1 < 3 (True) → (1, 3)
- 1 < 5 (True) → (1, 5)

2. For x = 2:
- 2 < 1 (False)
- 2 < 3 (True) → (2, 3)
- 2 < 5 (True) → (2, 5)

3. For x = 3:
- 3 < 1 (False)
- 3 < 3 (False)
- 3 < 5 (True) → (3, 5)

4. For x = 4:
- 4 < 1 (False)
- 4 < 3 (False)
- 4 < 5 (True) → (4, 5)

5. For x = 5:
- 5 < 1 (False)
- 5 < 3 (False)
- 5 < 5 (False)

6. For x = 6:
- 6 < 1 (False)
- 6 < 3 (False)
- 6 < 5 (False)

Now, we can compile the ordered pairs:
R = {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}

### Step 2: Plot R on a coordinate diagram of A × B.

To plot the relation R on a coordinate diagram, we will plot each ordered pair on a Cartesian plane where the x-axis represents elements from set A and the y-axis represents elements from set B.

- Plot (1, 3)
- Plot (1, 5)
- Plot (2, 3)
- Plot (2, 5)
- Plot (3, 5)
- Plot (4, 5)

### Step 3: State the domain, range, and codomain of R.

- **Domain**: The set of all first elements (x-values) from the ordered pairs in R.
- Domain = {1, 2, 3, 4}

- **Range**: The set of all second elements (y-values) from the ordered pairs in R.
- Range = {3, 5}

- **Codomain**: The set B from which the second elements are taken.
- Codomain = {1, 3, 5}

### Final Answers:
(i) R = {(1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)}
(ii) Points plotted on the coordinate diagram: (1, 3), (1, 5), (2, 3), (2, 5), (3, 5), (4, 5)
(iii) Domain = {1, 2, 3, 4}, Range = {3, 5}, Codomain = {1, 3, 5}