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Check whether the following relations ar...

Check whether the following relations are functions or not:
`R_(1)={(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)}`
` R_(2)={(2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7)}`
`R_(3)={(1,3),(1,5),(2,5)}`

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To determine whether the given relations are functions, we need to check if each input (first element of each ordered pair) is associated with exactly one output (second element of each ordered pair). ### Step-by-Step Solution: 1. **Check Relation R1:** - Relation: \( R_1 = \{(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)\} \) - Inputs (first elements): 2, 5, 8, 11, 14, 17 - Outputs (second elements): 1, 1, 1, 1, 1, 1 - Each input (2, 5, 8, 11, 14, 17) is associated with the output 1. - Since every input has exactly one output, **R1 is a function**. 2. **Check Relation R2:** - Relation: \( R_2 = \{(2,1), (4,2), (6,3), (8,4), (10,5), (12,6), (14,7)\} \) - Inputs: 2, 4, 6, 8, 10, 12, 14 - Outputs: 1, 2, 3, 4, 5, 6, 7 - Each input is associated with a unique output. - Since every input has exactly one output, **R2 is a function**. 3. **Check Relation R3:** - Relation: \( R_3 = \{(1,3), (1,5), (2,5)\} \) - Inputs: 1, 1, 2 - Outputs: 3, 5, 5 - The input 1 is associated with two different outputs (3 and 5). - Since the input 1 has more than one output, **R3 is not a function**. ### Summary: - **R1 is a function.** - **R2 is a function.** - **R3 is not a function.**
To determine whether the given relations are functions, we need to check if each input (first element of each ordered pair) is associated with exactly one output (second element of each ordered pair). ### Step-by-Step Solution: 1. **Check Relation R1:** - Relation: \( R_1 = \{(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)\} \) - Inputs (first elements): 2, 5, 8, 11, 14, 17 - Outputs (second elements): 1, 1, 1, 1, 1, 1 ...
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