For the following items, determine whether the relation is a function. Explain why or why not.
(a) {(a,b),(c,d),(e,d)}
(b) {(5,2),(6,1),(6,2),(4.8)}
(a) yes - all the 1st elements are unique
(b) no. How can f(6) be both 1 and 2?
To determine if a relation is a function, we need to check if there are any duplicate inputs with different outputs.
(a) {(a,b),(c,d),(e,d)}:
In this relation, each input appears only once. However, the output 'd' appears twice for different inputs 'c' and 'e'. Therefore, this relation is not a function.
(b) {(5,2),(6,1),(6,2),(4.8)}:
In this relation, we have two inputs '6' with different outputs '1' and '2'. The input '6' appears twice with different outputs, thus violating the definition of a function. Therefore, this relation is not a function.
In summary:
(a) {(a,b),(c,d),(e,d)} is not a function.
(b) {(5,2),(6,1),(6,2),(4.8)} is not a function.
To determine whether a relation is a function, we need to check if each input element (x-coordinate) in the relation corresponds to only one output element (y-coordinate). If there is any input element that has multiple output elements associated with it, then the relation is not a function.
Let's evaluate each item one by one:
(a) {(a,b),(c,d),(e,d)}
In this item, we have three input-output pairs: (a,b), (c,d), (e,d). Here, the input element 'd' is associated with two different output elements - 'b' and 'd'. Since 'd' has multiple outputs, this relation is not a function.
(b) {(5,2),(6,1),(6,2),(4.8)}
In this item, we have four input-output pairs: (5,2), (6,1), (6,2), (4.8). Here, the input element '6' is associated with two different output elements - '1' and '2'. Similar to the previous item, this relation is also not a function.
In conclusion, neither of the given relations is a function because at least one input element has multiple output elements associated with it.