INSA de ROUEN

2012/2013

Tutorial 6 : image, inverse image through a mapping. Internal binary operation ½ Exercise 1 : let f be the mapping

R x

→ 7→

R x 2 + 4x + 1

1. Show that f is a bijection from [−2; +∞[ into its image (precise it) and give its inverse. 2. Find f ([−3; 0], f −1 ({1}), f −1 ({−4}) et f −1 ([0; 1[). Exercise 2 : let n ∈ N∗ . Find the image of the mapping ( Exercise 3 : let f be the mapping

½

R∗+

→

R

x

7→

x ln(x) +

R∗+ x

→ 7 →

R x n ln(x)

1 . x

1. Find the greatest interval on which f is injective. ¶ µ 1 2. Find f ([1; e]) and f [ ; +∞[ . 2 ( Exercise 4 : let f be the mapping

R∗+

→

x

7→

R ³ ´ π Find f (]0; 1]) and f −1 ({0}). . sin x

Exercise 5 : let E and F be two sets, f : E → F a mapping, A and A 0 two subsets of E and B and B 0 two subsets of F . 1. Show f −1 (B ∪ B 0 ) = f −1 (B ) ∪ f −1 (B 0 ). 2. Show f −1 (B ∩ B 0 ) = f −1 (B ) ∩ f −1 (B 0 ). 3. Show f (A ∪ A 0 ) = f (A) ∪ f (A 0 ). 4. Show that f (A ∩ A 0 ) = f (A) ∩ f (A 0 ) is false but is true when f is injective. Exercise 6 : we define on Z the following internal binary operation : − : (a, b) 7→ a − b and ∗ : (a, b) 7→ a 2 + b 2 Study the associativity, the commutativity, the regularity of its elements, the existence of an identity element and the existence of inverse of elements for these operations. Exercise 7 : in E = [0; 1], we define the binary operation ∗ by x ∗ y = x + y − x y. 1. Check if ∗ is internal. 2. Study its properties (see above). 3. Same question for E =] − ∞; 1[. Exercise 8 : let ∆ be an equilateral triangle of center O. We consider the set I of the isometries keeping the triangle. (that is to say that through them the image of ∆ is ∆.) On I we use the composition operation. Study the properties of that operation on that set.