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Let f: X → Y and g : Y → Z be arbitrary maps of sets (a) Show that if f and g are injective then so is the composition go f (b) Show that if f and g are surjective then so is the composition go f (c) Show that if f and g are bijective then so is the composition gof and (gof)-¹ (d) Show that f: X → Y is injective iff there exists h: Y→ X such that h o f = (e) Show that f: X→Y is surjective iff there exists h: Y→ X such that foh: only if part requires the axiom of choice.

Let f: X → Y and g : Y → Z be arbitrary maps of sets (a) Show that if f and g are injective then so is the composition go f (b) Show that if f and g are surjective then so is the composition go f (c) Show that if f and g are bijective then so is the composition gof and (gof)-¹ (d) Show that f: X → Y is injective iff there exists h: Y→ X such that h o f = (e) Show that f: X→Y is surjective iff there exists h: Y→ X such that foh: only if part requires the axiom of choice.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5. Prove analogues of the results in problems 2 and 3 for arbitrary unions and intersections U₁4 and Ax
Transcribed Image Text:5. Prove analogues of the results in problems 2 and 3 for arbitrary unions and intersections U₁4 and Ax
All 2. Let f: X→ Y and g: Y→ Z be arbitrary maps of sets (a) Show that if f and g are injective then so is the composition gof (b) Show that if f and g are surjective then so is the composition gof (c) Show that if f and g are bijective then so is the composition gof and (gof)-¹ (d) Show that f: X→Y is injective iff there exists h: Y→X such that hof (e) Show that f: X→ Y is surjective iff there exists h: Y→ X such that foh only if part requires the axiom of choice. = 3. Show that it is always true that f(AUB) = f(A) U f(B) and f(An B) c f(A) r (a) Show, by example, that it might happen that f(An B) ‡ f(A)nf(B) (b) Show that f(An B) = f(A)n f(B) if f is injective. ||| H O
Transcribed Image Text:All 2. Let f: X→ Y and g: Y→ Z be arbitrary maps of sets (a) Show that if f and g are injective then so is the composition gof (b) Show that if f and g are surjective then so is the composition gof (c) Show that if f and g are bijective then so is the composition gof and (gof)-¹ (d) Show that f: X→Y is injective iff there exists h: Y→X such that hof (e) Show that f: X→ Y is surjective iff there exists h: Y→ X such that foh only if part requires the axiom of choice. = 3. Show that it is always true that f(AUB) = f(A) U f(B) and f(An B) c f(A) r (a) Show, by example, that it might happen that f(An B) ‡ f(A)nf(B) (b) Show that f(An B) = f(A)n f(B) if f is injective. ||| H O
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