Exercise 2 : let f be the mapping. R2. â. R3. (x, y) â (1,x â y, y) . Is it injective ? Surjective ? Exercise 3 : find all the injective mappings f : N â N such that ân ...
Tutorial 5 : sets, subsets, injection, surjection, bijection Exercise 1 : let A, B,C be three subsets of a set E . Prove the following equalities : • A ∩ (B ∪C ) = (A ∩ B ) ∪ (A ∩C ) • A ∪ (B ∩C ) = (A ∪ B ) ∩ (A ∪C ) • A ∪B = A ∩B • A ∩B = A ∪B Exercise 2 : let f be the mapping
R2 (x, y)
R3 . Is it injective ? Surjective ? (1, x − y, y)
→ 7 →
Exercise 3 : find all the injective mappings f : N → N such that ∀n ∈ N, f (n) ≤ n. Exercise 4 : let E , F,G be three sets, and f : E → F and g : F → G be two mappings. Show the following statements : • g o f is injective ⇒ f is injective. • g o f is surjective ⇒ g is surjective. z +i . z −i Show that f is a bijection from D = {z ∈ C/|z| < 1} into P = {z ∈ C/Re(z) < 0}.
Exercise 5 : for all z ∈ C\{i }, we define f (z) =
Exercise 6 : let E , F,G be three sets, and f : E → F and g : F → G be two mappings. Show that : • g o f is surjective and g is injective ⇒ f is surjective. • g o f is injective and f is surjective ⇒ g is injective. Exercise 7 : N → N . n 7→ 2n Is f injective ? Surjective ?
1. Let f :
N ( n if n is even n 7→ 2 2n if n is odd Is g injective ? Surjective ?
2. Let g :
N
→
.
3. Give f o g and g o f . Remarks ? Exercise 8 : find the subsets X and Y of R defined by : X = {x ∈ R ∃n ∈ N NON ((x ≥ 1/n) ⇒ (x > 1 − 1/n))} Y = {x ∈ R ∀y ∈ [0; 1], (x > y) ⇒ (x > 2y)} Exercise 9 : let f :
