Determine which of the following relation is a function?

Question

Given two set $ A = \{0, 2, 4, 6\}$ and $B = \{1, 3, 5, 7\}$, determine which of the following relation is a function?

$(a) \{(6, 3), (2, 1), (0, 3), (4, 5)\}$,

$(b) \{(2, 3), (4, 7), (0, 1), (6, 5)\}$,

$(c) \{(2, 1), (4, 5), (6, 3)\}$,

$(d) \{(6, 1), (0, 3), (4, 1), (0, 7), (2, 5)\}$.

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My attempt :

"So far I know that for an equation to be functional, any x value cannot be repeated. You cannot have two x's. Y can repeat however much it wants to but despite that not being able to determine which are functional.

Answer 1

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

So, according definition of function, given relations with domain $A = \{0, 2, 4, 6\}$,

$(a) \{(6, 3), (2, 1), (0, 3), (4, 5)\} = \{ (0, 3),(2, 1), (4, 5),(6, 3)\}$ is function , since it defines for every element of function $(i.e.\{0, 2, 4, 6\})$ and has exactly one output for each element.

$(b) \{(2, 3), (4, 7), (0, 1), (6, 5)\} = \{(0, 1),(2, 3), (4, 7), (6, 5)\}$ is function , since it defines for every element of function $(i.e.\{0, 2, 4, 6\})$ and has exactly one output for each element.

$(c) \{(2, 1), (4, 5), (6, 3)\}$ is not a function, since it defines on element $0$ of element of function $(i.e.\{0, 2, 4, 6\})$.

$(d) \{(6, 1), (0, 3), (4, 1), (0, 7), (2, 5)\} = \{(0, 3),(0, 7),(2, 5),(4, 1),(6, 1) \}$ is not a function, since it not has exactly one output of element of function $(i.e.\{0, 2, 4, 6\})$. Element $0$ have two output $3$ and $7$.

Reference@Wikipedia.

Answer 2

A relation is a set of ordered pairs, which maps from one set, called the domain, to another set, called the co-domain.   Here $A$ is the domain, and $B$ is the co-domain.   All the sets given in the OP are indeed relations of $A\to B$; but are they functions?

A function is a relation where any element of the domain is mapped to at most one element in the co-domain.   That is: no two distinct pairs of the relation will share the same left-member (or "x"-value; or rather "A"-value in this case).

( PS: "at most one" means either one or none, but never two or more. )

That is it!   That's the only property you have to test.   There is no restriction on sharing the right members ("y"-values; or "B" values here).