what is the number of relation a=(0,1,2) on a?
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This article examines the concepts of a function and a relation.
A relation is any association or link between elements of one set, called the domain or (less formally) the set of inputs, and another set, called the range or set of outputs. Some people mistakenly refer to the range as the codomain(range), but as we will see, that really means the set of all possible outputs—even values that the relation does not actually use. (Beware: some authors do not use the term codomain(range), and use the term range instead for this purpose. Those authors use the term image for what we are calling range. So while it is a mistake to refer to the range or image as the codomain(range), it is not necessarily a mistake to refer to codomain as range.)
For example, if the domain is a set Fruits = {apples, oranges, bananas} and the codomain(range) is a set Flavors = {sweetness, tartness, bitterness}, the flavors of these fruits form a relation: we might say that apples are related to (or associated with) both sweetness and tartness, while oranges are related to tartness only and bananas to sweetness only. (We might disagree somewhat, but that is irrelevant to the topic of this book.) Notice that "bitterness", although it is one of the possible Flavors (codomain)(range), is not really used for any of these relationships; so it is not part of the range(or image) {sweetness, tartness}.
Another way of looking at this is to say that a relation is a subset of ordered pairs drawn from the set of all possible ordered pairs (of elements of two other sets, which we normally refer to as the Cartesian product of those sets). Formally, R is a relation if
{\displaystyle R\subseteq X\times Y=\{(x,y)|x\in X,y\in Y\}}for the domain X and codomain(range) Y. The inverse relation of R, which is written as R-1, is what we get when we interchange the X and Y values:
{\displaystyle R^{-1}=\{(y,x)|(x,y)\in R\}}Using the example above, we can write the relation in set notation: {(apples, sweetness), (apples, tartness), (oranges, tartness), (bananas, sweetness)}. The inverse relation, which we could describe as "fruits of a given flavor", is {(sweetness, apples), (sweetness, bananas), (tartness, apples), (tartness, oranges)}. (Here, as elsewhere, the order of elements in a set has no significance.)
One important kind of relation is the function. A function is a relation that has exactly one output for every possible input in the domain. (The domain does not necessarily have to include all possible objects of a given type. In fact, we sometimes intentionally use a restricted domain in order to satisfy some desirable property.) The relations discussed above (flavors of fruits and fruits of a given flavor) are not functions: the first has two possible outputs for the input "apples" (sweetness and tartness); and the second has two outputs for both "sweetness" (apples and bananas) and "tartness" (apples and oranges).
The main reason for not allowing multiple outputs with the same input is that it lets us apply the same function to different forms of the same thing without changing their equivalence. That is, if f is a function with a (or b) in its domain, then a = b implies that f(a) = f(b). For example, z - 3 = 5 implies that z = 8 because f(x) = x + 3 is a function defined for all numbers x.
The converse, that f(a) = f(b) implies a = b, is not always true. When it is, there is never more than one input x for a certain output y = f(x). This is the same as the definition of function, but with the roles of X and Y interchanged; so it means the inverse relation f-1 must also be a function. In general—regardless of whether or not the original relation was a function—the inverse relation will sometimes be a function, and sometimes not.
When f and f-1 are both functions, they are called one-to-one, injective, or invertible functions. This is one of two very important properties a function f might (or might not) have; the other property is called onto or surjective, which means, for any y ∈ Y (in the codomain), there is some x ∈ X (in the domain) such that f(x) = y. In other words, a surjective function f maps onto every possible output at least once.
A function can be neither one-to-one nor onto, both one-to-one and onto (in which case it is also called bijective or a one-to-one correspondence), or just one and not the other. (As an example which is neither, consider f = {(0,2), (1,2)}. It is a function, since there is only one y value for each x value; but there is more than one input x for the output y = 2; and it clearly does not "map onto" all integers.)