To determine whether the given relations are functions, we need to check if each input (first element of the ordered pair) has a unique output (second element of the ordered pair). A relation is a function if every element in the domain maps to exactly one element in the range.
Let's analyze the two relations step by step:
### Relation (i): `{(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}`
1. **Identify the ordered pairs**: The pairs are (2, 1), (5, 1), (8, 1), (11, 1), (14, 1), and (17, 1).
2. **Check the first elements**: The first elements are 2, 5, 8, 11, 14, and 17.
3. **Check the second elements**: The second element for all pairs is 1.
4. **Determine uniqueness**: Each first element (2, 5, 8, 11, 14, 17) maps to the same second element (1). There are no repeated first elements with different second elements.
5. **Conclusion**: Since every input has a unique output, this relation is a function.
### Domain and Range for Relation (i):
- **Domain**: {2, 5, 8, 11, 14, 17}
- **Range**: {1}
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### Relation (ii): `{(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}`
1. **Identify the ordered pairs**: The pairs are (2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), and (14, 7).
2. **Check the first elements**: The first elements are 2, 4, 6, 8, 10, 12, and 14.
3. **Check the second elements**: The second elements are 1, 2, 3, 4, 5, 6, and 7.
4. **Determine uniqueness**: Each first element maps to a different second element. There are no repeated first elements with different second elements.
5. **Conclusion**: Since every input has a unique output, this relation is also a function.
### Domain and Range for Relation (ii):
- **Domain**: {2, 4, 6, 8, 10, 12, 14}
- **Range**: {1, 2, 3, 4, 5, 6, 7}
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### Summary:
- Relation (i) is a function with Domain: {2, 5, 8, 11, 14, 17} and Range: {1}.
- Relation (ii) is a function with Domain: {2, 4, 6, 8, 10, 12, 14} and Range: {1, 2, 3, 4, 5, 6, 7}.
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