Y. The mapping f, which maps each element of X into an
element of Y, induces the following two important set mappings:
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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 A1, A2, .... , An 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 A1, A2, .... , An 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 [Ac] = (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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