The relation of a function

328. Consider the functions fromS ={−2,−1,0,1,2} toT ={1,2,3,4,5} defined by f(x) =x+ 3, and g(x) =x5−5x3+ 5x+ 3. Write down the set of ordered pairs (x, f(x)) forx∈Sand the set of ordered pairs (x, g(x)) for x∈S. Are the two functions the same or different? Problem 328 points out how two functions which appear to be different are actually the same on some domain of interest to us. Most of the time when we are thinking about functions it is fine to think of a function casually as a relationship between two sets. In Problem 328 the set of ordered pairs you wrote down for each function is called therelationof the function. When we want to distinguish between the casual and the careful in talking about relationships, our casual term will be “relationship” and our careful term will be “relation.” So relation is a technical word in mathematics, and as such it has a technical definition. Arelation from a setS to a set T is a set of ordered pairs whose first elements are in S and whose second elements are in T. Another way to say this is that arelation fromS toT is a subset of S×T.

A typical way to define afunction f from a set S, called thedomainof the function, to a set T, called the range, is thatf is a relationship fromS toT that relates one and only one member ofT to each element of X. We use f(x) to stand for the element of T that is related to the element x of S. If we wanted to make our definition more precise, we could substitute the word “relation” for the word “relationship” and we would have a more

precise definition. For our purposes, you can choose whichever definition you prefer. However, in any case, there is a relation associated with each function. As we said above, the relation of a function f : S → T (which is the standard shorthand for “f is a function fromS toT” and is usually read as f maps S to T) is the set of all ordered pairs (x, f(x)) such thatx is inS.

329. Here are some questions that will help you get used to the formal idea of a relation and the related formal idea of a function. S will stand for a finite set of size sand T will stand for a finite set of size t.

(a) What is the size of the largest relation fromS toT? (b) What is the size of the smallest relation fromS toT?

(c) The relation of a function f : S → T is the set of all ordered pairs (x, f(x)) withx ∈S. What is the size of the relation of a function from S to T? That is, how many ordered pairs are in the relation of a function from S toT? Online hint.

(d) We say f is a one-to-one1 function or injection from S to T if each member of S is related to a different element of T. How many different elements must appear as second elements of the ordered pairs in the relation of a one-to-one function (injection) from S toT?

(e) A function f :S → T is called an onto function orsurjection if each element of T isf(x) for somex∈S. What is the minimum size that S can have if there is a surjection from S toT?

330. When f is a function from S toT, the sets S and T play a big role in determining whether a function is one-to-one or onto (as defined in Problem 329). For the remainder of this problem, let S and T stand for the set of nonnegative real numbers.

(a) If f :S →T is given by f(x) =x2, isf one-to-one? Isf onto? (b) Now assume for the rest of the problem thatS0is the set of all real

numbers and g:S0 → T is given by g(x) = x2. Is g one-to-one? Is g onto?

1_{The phrase one-to-one is sometimes easier to understand when one compares it to the}

phrase many-to-one. John Fraliegh, an author of popular textbooks in abstract and linear algebra, suggests that two-to-two might be a better name that one-to-one.

A.1. RELATIONS AS SETS OF ORDERED PAIRS 149 (c) Assume for the rest of the problem that T0 is the set of all real numbers andh :S →T0 is given byh(x) =x2. Ish one-to-one? Is h onto?

(d) And if the function j : S0 → T0 is given by j(x) = x2, is j one-to-one? Is j onto?

331. If f :S →T is a function, we say that fmaps x toy as another way to say that f(x) = y. Suppose S = T = {1,2,3}. Give a function from S to T that is not onto. Notice that two different members of S have mapped to the same element of T. Thus when we say that f associates one and only one element of T to each element of S, it is quite possible that the one and only one element f(1) thatf maps 1 to is exactly the same as the one and only one element f(2) that f maps 2 to.