Problem 1.1E: Let f:AB. Let A0AandB0B. Show that A0f1(f(A0)) and that equality holds if f is injective. Problem 1.2E Problem 2.1E: Let f:AB and let AiAandBiBfori=0andi=1. Show that f1 preserves inclusions, unions, intersections,... Problem 2.2E: Let f:AB and let AiAandBiBfori=0andi=1. Show that f1 preserves inclusions, unions, intersections,... Problem 2.3E: Let f:AB and let AiAandBiBfori=0andi=1. Show that f1 preserves inclusions, unions, intersections,... Problem 2.4E: Let f:AB and let AiAandBiBfori=0andi=1. Show that f1 preserves inclusions, unions, intersections,... Problem 2.5E Problem 2.6E: Let f:AB and let AiAandBiBfori=0andi=1. Show that f1 preserves inclusions, unions, intersections,... Problem 2.7E: Let f:AB and let AiAandBiBfori=0andi=1. Show that f1 preserves inclusions, unions, intersections,... Problem 2.8E: Let f:AB and let AiAandBiBfori=0andi=1. Show that f1 preserves inclusions, unions, intersections,... Problem 3.1E: Show that b, c, f, and g of Exercise 2 hold for arbitrary unions and intersections. Let f:AB and let... Problem 3.2E: Show that b, c, f, and g of Exercise 2 hold for arbitrary unions and intersections. Let f:AB and let... Problem 3.3E: Show that b, c, f, and g of Exercise 2 hold for arbitrary unions and intersections. Let f:AB and let... Problem 3.4E: Show that b, c, f, and g of Exercise 2 hold for arbitrary unions and intersections. Let f:AB and let... Problem 4.1E: Let f:AB and g:BC. If C0C, show that (gf)1(C0)=f1(g1(C0)). Problem 4.2E: Let f:AB and g:BC. If f and g are injective, show that gf is injective. Problem 4.3E: Let f:AB and g:BC. If gf is injective, what can you say about injectivity of f and g? Problem 4.4E: Let f:AB and g:BC. If f and g are surjective, show that gf is surjective. Problem 4.5E: Let f:AB and g:BC. If gf is surjective, what can you say about surjectivity of f and g? Problem 4.6E: Let f:AB and g:BC. Summarize your answers to b-e in the form of a theorem. Problem 5.1E: In general, let us denote the identity function for a set C by iC. That is, define iC:CC to be the... Problem 5.2E: In general, let us denote the identity function for a set C by iC. That is, define iC:CC to be the... Problem 5.3E: In general, let us denote the identity function for a set C by iC. That is, define iC:CC to be the... Problem 5.4E: In general, let us denote the identity function for a set C by iC. That is, define iC:CC to be the... Problem 5.5E: In general, let us denote the identity function for a set C by iC. That is, define iC:CC to be the... Problem 6E: Let f: be the function f(x)=x3x. By restricting the domain and range of f appropriately, obtain from... format_list_bulleted