the inverse image commutes with set operations
Theorem.
Let be a mapping from to . If is
a (possibly uncountable) collection of subsets in , then
the following relations
hold for the inverse image
:
-
(1)
-
(2)
If and are subsets in , then we also have:
-
(3)
For the set complement
,
-
(4)
For the set difference
,
-
(5)
For the symmetric difference
,
Proof. For part (1), we have
Similarly, for part (2), we have
For the set complement, suppose . This is equivalent to
, or , which is equivalent to
. Since the set difference can be
written as , part (4) follows from parts (2) and
(3). Similarly, since ,
part (5) follows from parts (1) and
(4).
Title | the inverse image commutes with set operations |
---|---|
Canonical name | TheInverseImageCommutesWithSetOperations |
Date of creation | 2013-03-22 13:35:24 |
Last modified on | 2013-03-22 13:35:24 |
Owner | matte (1858) |
Last modified by | matte (1858) |
Numerical id | 11 |
Author | matte (1858) |
Entry type | Proof |
Classification | msc 03E20 |
Related topic | SetDifference |