Reference: Topology and Modern Analysis, G F Simmons, Tata McGraw Hill Publications, India.
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
This is by definition of product and the fact that the co-ordinates are ordered and the fact that , , , and .
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 .
From question IIb above, the difference of any two pairs of sets is also in .