To determine which of the given relations are functions, we need to verify if each relation maps every input to exactly one unique output. In simpler terms, for a set of ordered pairs to represent a function, each first element (input) must be associated with exactly one second element (output).
Let's go through each relation step-by-step:
### Relation [tex]\( A = \{(1, 2), (2, 3), (3, 4), (4, 5)\} \)[/tex]
- Inputs: 1, 2, 3, 4
- Outputs: 2, 3, 4, 5
In relation A, each input value is unique and maps to exactly one output value. Therefore, [tex]\( A \)[/tex] is a function.
### Relation [tex]\( B = \{(3, 3), (4, 4), (5, 5), (6, 6)\} \)[/tex]
- Inputs: 3, 4, 5, 6
- Outputs: 3, 4, 5, 6
In relation B, each input value is unique and maps to exactly one output value. Therefore, [tex]\( B \)[/tex] is a function.
### Relation [tex]\( C = \{(1, 0), (0, 1), (-1, 0), (0, -1)\} \)[/tex]
- Inputs: 1, 0, -1, 0
- Outputs: 0, 1, 0, -1
In relation C, the input value 0 appears twice (in the pairs [tex]\((0, 1)\)[/tex] and [tex]\((0, -1)\)[/tex]) and maps to different outputs (1 and -1). Therefore, [tex]\( C \)[/tex] is not a function.
### Relation [tex]\( D = \{(a, b), (b, c), (c, d), (a, d)\} \)[/tex]
- Inputs: a, b, c, a
- Outputs: b, c, d, d
In relation D, the input value [tex]\(a\)[/tex] appears twice (in the pairs [tex]\((a, b)\)[/tex] and [tex]\((a, d)\)[/tex]) and maps to different outputs (b and d). Therefore, [tex]\( D \)[/tex] is not a function.
### Conclusion
From our analysis, the relations that are functions are:
- Relation [tex]\( A \)[/tex]
- Relation [tex]\( B \)[/tex]
Thus, the functions are [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
