Reference: Topology and Modern Analysis, G F Simmons, Tata McGraw Hill Publications, India.
Problems:
I) The graph of a mapping is a subset of the product
. What properties characterize the graphs of mappings among all subsets of
?
Solution I: composition.
II) Let X and Y be non-empty sets. If and
are subsets of X, and
and
are subsets of Y, then prove the following
(a)
(b)
Solution IIa:
TPT:
This is by definition of product and the fact that the co-ordinates are ordered and the fact that ,
,
, and
.
Solution IIb:
Let , but
. So, the element may belong to
or it could happen that it belongs to
, but to
(here we need to remember that the element is ordered); so, also it could be the other way: it could belong to
but to
also. The same arguments applied in reverse establish the other subset inequality. Hence, done.
III) Let X and Y be non-empty sets, and let A and B be rings of subsets of X and Y, respectively. Show that the class of all finite unions of sets of the form with
and
is a ring of subsets of
.
Solution III:
.
From question IIb above, the difference of any two pairs of sets is also in .
Hence, done.
Regards,
Nalin Pithwa