Let X be a non-empty set and f a mapping of X into itself. Show that f is one-to-one onto iff there exists a mapping g of X into itself such that fg = gf = iX. If there exists a mapping g with this property, then there is only one such mapping. Why?

Topology Sets and Functions (XXXIX) Functions Let X be a non-empty set and f a mapping of X into itself. Show that f is one-to-one onto iff there exists a mapping g of X into itself such that fg = gf = iX. If there exists a mapping g with this property, then there is only one such mapping. Why? See the attached file.

The graph of a mapping f:X→Y is a subset of the product X×Y. What properties characterize the graphs of mappings among all subsets of X×Y?

Topology Sets and Functions (XL) Functions The graph of a mapping f:X→Y is a subset of the product X×Y. What properties characterize the graphs of mappings among all subsets of X×Y? See the attached file.

Let X and Y be non-empty sets. If A1 and A2 are subsets of X, and B1 and B2 subsets of Y. Show that (A1×B1)∩(A2×B2) = (A1∩A2)×(B1∩B2).

Topology Sets and Functions (XLI) Functions Let X and Y be non-empty sets. If A1 and A2 are subsets of X, and B1 and B2 subsets of Y. Show that (A1×B1)∩(A2×B2) = (A1∩A2)×(B1∩B2). See the attached file.

Let X and Y be non-empty sets. If A1 and A2 are subsets of X, and B1 and B2 subsets of Y. Show that (A1×B1) – (A2×B2) = (A1 – A2)×(B1 – B2) U(A1∩A2)×(B1 – B2) U(A1 – A2)×(B1∩B2)

Topology Sets and Functions (XLII) Functions Let X and Y be non-empty sets. If A1 and A2 are subsets of X, and B1 and B2 subsets of Y. Show that (A1×B1) – (A2×B2) = (A1 – A2)×(B1 – B2) U(A1∩A2)×(B1 – B2) U(A1 – A2)×(B1∩B2) See the attached file.

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 A×B with AЄA and BЄB is a ring of subsets of X×Y.

Topology Sets and Functions (XLIII) Functions 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 A×B with AЄA and BЄB is a ring of subsets of X×Y. See the attached file.