Meaning of function or mapping or transformation in brief words

DEFINITION:

Given two sets X and Y, a transformation (also called a function or mapping) f: X \rightarrow Y of X into Y is a triple (X,Y,G) where G itself is a collection of ordered pairs (x,y), the first element of each pair being an element of X, and the second an element of Y, with the condition that each element of X appears as the first element of exactly one pair of G.

If each element of Y appears as the second element of some pair in G, then the transformation is said to be onto.

If each element of Y which appears at all, appears as the second element of exactly one pair in G, then f is said to be one-to-one. Note that a transformation can be onto without being one-to-one and conversely.

As an aid in understanding the above definition, consider the equation y=x^{2} where x is a real number. We may take X to be the set of all real numbers and then the collection G is the set of pairs of (x,x^{2}). Taking Y to be just the set of nonnegative reals will cause f to be onto. But if Y is all real numbers, or all reals greater than -7, or any other set containing the nonnegative reals as a proper subset, the transformation is not onto. With each new choice of Y, we change the triple and hence the transformation.

Continuing with the same example, we could assume that X is the set of nonnegative reals also. Then the transformation is one-to-one, as is easily seen. Depending upon the the choice of Y, the transformation may or may not be onto, of course. Thus, we see that we have stated explicitly the conditions usually left implicit in defining a function in analysis. The reader will find that the seemingly pedantic distinctions made here are really quite necessary.

If f: X \rightarrow Y is a transformation of X into Y, and x is an element of the set X, then we let f(x) denote the second element of the pair in G whose first element is x. That is, f(x) is the “functional value” in Y of the point x. Similarly, if Z is a subset of X, then f(Z) denotes that subset of Y composed of all points f(z), where z is a point in Z. If y is a point of Y, then by f^{-1}(y) is meant the set of all points in X for which f(x)=y; and if W is a subset of Y, then f^{-1}(W) is the set theoretic union of the sets $f^{-1}(w)$, where w is in W. Note that f^{-1} can be used as a symbol to denote the triple (X,Y,G^{'}) wjere G^{'} consists of all pairs (y,x) that are reversals of pairs in G. But f^{-1} is a transformation only if f is both one-to-one and onto. If A is a subset of X and if f: X \rightarrow Y, then f may be restricted to A to yield a transformation denoted by f|A: A \rightarrow Y, and called the restriction of f to A.

Cheers,

Nalin Pithwa

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