Question

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Answer 1

D. It cannot be a function because two of the ordered pairs have the same first number.

Answer 2

d. Given sets X and Y, a function from X to Y is a set of ordered pairs F of members of these sets such that for every x in X there is a unique y in Y for which the pair (x,y) is in F.An example of a function from the reals to the reals is given by the set of ordered pairs (x, x2), where x is a real number. This squaring function from the reals to the reals is not considered the same as the function from the reals to the non-negative reals as they are two different types of entities.
The above definition of "a function from X to Y" is generally agreed on, however there are two different ways a "function" is normally defined where the domain X and codomain Y are not explicitly or implicitly specified. Usually this is not a problem as the domain and codomain normally will be known. With one definition saying the function defined by f(x) = x2 on the reals does not completely specify a function as the codomain is not specified, and in the other it is a valid definition.
In one definition a function is an ordered triple of sets, written (X, Y, F), where X is the domain, Y is the codomain, and F is a set of ordered pairs (x, y). In each of the ordered pairs, the first element x is from the domain, the second element y is from the codomain, and a necessary condition is that every element in the domain is the first element in exactly one ordered pair.
In the other definition a function is defined as a set of ordered pairs where each first element only occurs once. The domain is the set of all the first elements of a pair and there is no explicit codomain separate from the image. Concepts like surjective don't apply to such functions, a codomain must be explicitly specified.
Functions are commonly defined as a type of relation. A relation from X to Y is a set of ordered pairs (x, y) with and . A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. However when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a triple (X, Y, F), however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. The definition of a function as a triple defines a function as a type of correspondence, whereas the definition of a function as an ordered pair defines a function as a type of relation.

Answer 3

-8 is paired with both 1 and -6. The relaton cannot be function because there is not a unique y for a given x.

D) pictures the mapping and apty describes it as "not a function."