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

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