Example:
If S is some set, let f be a function between S and P(S), i.e. if
s 
S, then
f(s) P(S).
Define the set
X = {s S :
{s} 
f(s) = 0} or, in other words
if T = f(t), then t is in X if and only if t is not
contained in T. What is the set X in the following example ?
Let S = {1, 2, 3}. Then P(S) = { 0, {1}, {2}, {3},
{1,2}, {1,3}, {2,3}, {1,2,3} }. Define some function f via
- 1 is associated with {3}
- 2 is associated with {1,2,3}
- 3 is associated with {1,2}.
Then the set X defined above consist of the following elements:
- f(1) = {3}, and 1 is not in {3}. Thus, {1} is in X.
- f(2) = {1,2,3}, and 2 is in {1,2,3}. Thus, {2} is not in X.
- f(3) = {1, 2}, and 3 is not in {1,2}. Thus, {3} is in X.
Hence, X = {1, 3}.
Incidently, there is no element from S that is mapped to the
set X.
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