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Functions in set theory

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Those who have taken algebra ought to be familiar with the concept of a 'function.' This concept has quite an important use in set theory. "A function f from set X to set Y is a relation in which each member of X has a unique partner in Y"(Steinhart, p. 49). Each element from X has exactly one corresponding member in y. Functions are symbolized as:

f: X ---> Y

"If a function f from X to Y pairs off some x in X with some y in Y, we say f of x is y, and we write this f(x)=y. Thus for any x in X, the symbolism f(x) refers to the unique partner of x in Y"(Steinhart, p. 49).

For example, using a function, we can "assign" each man of X a wife in Y. X is known as the domain of f whereas Y is known as the codomain of f. "We say function f maps its domain to its codomain or that it is a map from its domain to its codomain"(Steinhart, p. 50). Keep in mind that this pairing is from only one member of X with only one member of Y. "This is why functions are singled out for special focus. they are unambiguous relations. A seating assignment function does not pair a student with more than one desk. The seating assignment uniquely determines your seat. There is no ambiguity or confusion about your seat. A grading function does not pair a student with more than one grade. The grading function uniquely determines your grade; there is no ambiguity"(Steinhart, p. 50).

Every member of X must have only one counterpart in Y, but the opposite is not true. "Although a function from X to Y cannot associate one member of X with many members of Y, it can associate one member of Y with many members of X. One student cannot get many grades, but one grade can be given to many students"(Steinhart, p. 50). Of coursre, there are one-to-one functions. These occur when the function

"never assigns one member of Y to more than one member of X (though it may leave some members of Y unassigned). For example, the seating assignment function is one-to-one. One estudent gets one desk, and one desk gets one student (though if there are more desks than students, some desks may get no students). Since the seating assignment is one-to-one, if the desk assigned to Superman = the desk assigned to Clark Kent, then you can infer that Superman = Clark Kent. But the grading function need not be one-to-one. One estudent gets one grade, but one grade can be given to many students"(Steinhart, p. 50).

Stated formally, "a function f is one-to-one if, and only if, f(x) = f(y) implies x = y"(Steinhart, p. 51).

The Superman/Clark Kent analogy for one-to-one functions ought to come to mind here. Because the seating assignment is one-to-one, if a desk is assigned to one, we can infer that it is identical with the other. A seat assigned to Clark Kent is necessarily a seat assigned to superman.

We speak of one-to-one mappings in terms of a function of X onto Y. That is, "onto" as opposed to "into." The latter does not refer to one-to-one mappings. Attention to this variation in preposition is very important in set theory, as it is easy to fail to notice this subtle but important difference in phraseology.

"A function from X to Y is onto if and only if it associates every member of Y with some member of X. No members of Y are left without partners in X. For example, if there are exactly as many desks in a classroom as there are estudents in that class, then every student getts one desk and every desk gets one student. Hence the relation f that partners students with desks is a function from Students onto Desks. The fact that every student gets one desk makes f a function; the fact that every desk gets one student makes f onto. More formally, a function f is onto iff for every y ∈ Y, there is some x ∈ X such that f(x)=y. A function t hat is not onto is into. For example, if there are more desks than students ins ome class, then the seating assignment is a function from the set of students into the set of desks"(Steinhart, p. 51).

Steinhart articulates the technical terminology for these different kinds of functions:

"A one-to-one function is sometimes said to be an injection. An onto function is sometimes said to be a surjection. And a function that is both one-to-one and onto is sometimes called a bijection. It is also sometimes known by the term 1-1 correspondence"(Steinhart, p. 51).

We formally define the inverse of a function as "f^-1 = {(y, x) | (x, y) ∈ f}."

Let's look at the concept of a "characteristic function." It is defined as "a function f from some set S onto {0, 1}. A characteristic function over a set S is a way of specifying a subset of S. For any x in S, x is in the subset specified by f if f(x) = 1 and x is not in that subset if f(x) = 0"(Steinhart, p. 54).

Let's look at an example of what this means, using the examples provided by Steinhart's book. Suppose we have a set {A, E, I, O, U, Y}. Suppose we articulate this in terms of a function C, which equals {(A, 0), (E, 1), (I, 1), (O, 1), (U, 0), (Y, 0)}. This function is an example of a "characteristic function." The reason is because it specifies subsets of the initial set. "We can use characteristic functions to introduce the idea of a set of functions. The set of characteristic functions over a set S is the set of all f such that f: S ---> {0, 1}. In symbols, the set of characteristic functions over S = {f|f: S ---> {0, 1}}"(Steinhart, pp. 54-55).

Let's look at isomorphisms. Formally defined, an isomorphism is a "structure-preserving bijection"(Steinhart, p. 55). Put simply, this means that "it is a 1-1 correspondence between two systems that exactly translates the structure of the one into the structure of the other. Thus two systems that are isomorphic have the same structure or form"(Steinhart, p. 55).

Let's look at an example Steinhart gives us:

theSeasons = {winter, spring, summer, fall};

theDirections = {north, south, east, west}.

Note that there is a one-to-one correspondence between the seasons and the geographical regions in which they occur. The correspondence is thus an instance of structure-preserving bijection, or isomorphism. Steinhart notes that there are two structural orientations the two sets have with one another. Orthogonality and opposite. While North and South are opposite, North and West are orthogonal(Steinhart, p. 55). Represented formally, we can write this f: theSeasons --> theDirections.

We can use functional notation to express the sum of a quality of the members of a set (Steinhart, p. 59). Steinhart's example is that of a set of physical things. We can express the sum of the weight of these things by writing:

"the sum, for all x in A, of the weight of x =

After the equals sign, Steinhart has a large sigma, under which is the statement x ∈ A, and to the right of which is WeightOf(x). The reader would do well to copy this onto his own paper to see what this looks like, as the exact notation with which Steinhart represents the concept is too difficult to fit into this format. We can use such notation to represent nested sums, which are sums within sums:

"Nested sums are useful in formalization of utilitarianism. For instance, there are many persons in a world; each person is divisible into instantaneous person-stages; each stage has some degree of pleasure. Suppose, crudely, that utility is just total pleasure (total hedonic value). The hedonic value of a person is the esum, for all his or her stages, of the pleasure of that stage. The hedonic value of a world is the sum, for every person in the world, of the hedonic value of the person. Thus we have a nested sum"(Steinhart, p. 60).

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.