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Set functions

Consider an arbitrary mapping f: X Y. The mapping f, which maps each element of X into an element of Y, induces the following two important set mappings:

1. Forward set mapping. If A is a subset of X, then its image f [A] is the subset of Y defined by

f [A] = {f(x): x ε A}.

Example 1. In Fig. 1 the mapping f maps the set {a, b} into the set {2, 3}.

The forward set mapping is the mapping that maps each subset A of X into its image f [A] in Y.

2. Backward set mapping. If B is a subset of Y, then its inverse
image f ^{-1}[B] is the subset of X defined by

f ^{-1}[B] = {x: f(x) ε B]

i.e. those elements in A that map into B.

Example 2. In Fig. 1 the mapping f ^{-1 }maps the set {2, 3} into the set {a, b, c, d}

The backward set mapping is the mapping that maps every subset B in Y back into that set of elements in X that map into it.

In other words, a function f: X
Y induces a function, also denoted by f, from the power set
P(X) of X (i.e. the collection of all subsets of X) into the power set P(Y) of Y, and a function f ^{-1}
from P(Y) into P(X). The induced functions f and f ^{-1} are called set functions because they
map sets into sets.

Note that we are using different brackets to distinguish between a function and its associated set function i.e. f(a) denotes a value of the original function and f [A] and f [B] denote values of the associated set functions.

The forward and backward set mappings possess various properties. They are:

Properties of forward set mappings. Let f: X Y . Then for any subsets A and B of X

and, more generally, for any collection of sets A_{1}, A_{2}, .... , A_{n} of X

Example 3. Let us use the mapping of Fig. 1 to confirm formulas 4) and 6). Let A = {a, b} and B = {c, d}. Now substitute A and B into formulas 4) and 6) to confirm them.

Formula 4):

A - B = {a, b}

f [A - B] = f [{a, b}] = {2, 3}

f [A] = {2, 3}; f [B] = {3}

f [A] - f [B] = {3}

Formula 6):

f {A B] =

f [A] f [B] = {3}

Properties of backward set mappings. Let f: X Y . Then for any subsets A and B of X

and, more generally, for any collection of sets A_{1}, A_{2}, .... , A_{n} of X

Example 4. Let us use the mapping of Fig. 1 to confirm formula 6). Let A = {2, 3} and B = {1, 2}.

f ^{-1}[A
B] = f ^{-1}[{2}] = {a}

f ^{-1}[A] = {a, b, d, e}

f ^{-1}[B] = {a, c}

f ^{-1}[A]
f ^{-1}[ B] = {a}

Theorem 1. Let f: X
Y and let A
Y. Then f ^{-1 }[A^{c}] = (f ^{-1 }[A])^{c}.

Theorem 2. Let f: X Y and let A X and B Y. Then:

References.

1. Lipschutz. General Topology.

2. Simmons. Introduction to Topology and Modern Analysis.

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