MAS111_09_assignment_13_1

# MAS111_09_assignment_13_1 - that g ◦ f = 1 A . Prove that...

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MAS 111 FOUNDATION OF MATHEMATICS PRACTICE EXERCISES FUNCTIONS 1. Does the formula f ( x ) = 1 x 2 - 2 deﬁne a function f : R -→ R ? 2. For each of the following functions, determine whether it is one-to-one and determine its range. (a) f : Z -→ Z , f ( x ) = 2 x + 1; (b) g : Q -→ Q , g ( x ) = 2 x + 1; (c) h : Z -→ Z , h ( x ) = x 3 - x ; (d) k : R -→ R , k ( x ) = e x ; (e) l : [ - π/ 2 ,π/ 2] -→ R , l ( x ) = sin x ; (f) j : [0 ] -→ R , j ( x ) = sin x . 3. For each of the following functions g : R -→ R , determine whether the function is one-to-one and whether it is onto. If the function is not onto, determine the range g ( R ). (a) f ( x ) = x + 7; (b) g ( x ) = 2 x - 3; (c) h ( x ) = - x + 5; (d) k ( x ) = x 2 ; (e) l ( x ) = x 2 + x ; (f) j ( x ) = x 3 . 4. Let A = { x R : - 1 < x < 1 } . Show that the function R -→ A deﬁned by f ( x ) = x 1 + | x | is a one-to-one and onto function. 1

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5. Let α be an irrational number. Deﬁne M = Q ∪{ α } and let f : M -→ M be deﬁned as f ( x ) = 1 - x 1 + x if x Q / {- 1 } , - 1 if x = α, α if x = - 1 . Show that f is a bijection. 6. The following two statements justify the deﬁnition of the inverse func- tion. We can conclude that only bijections can have inverses. (a) Suppose that f : A -→ B and g : B -→ A are functions such
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Unformatted text preview: that g ◦ f = 1 A . Prove that f is one-to-one. (b) Suppose that f : A-→ B and g : B-→ A are functions such that f ◦ g = 1 B . Prove that f is an onto function. 7. Let f : A-→ B be an onto function. Prove that there is a function g : B-→ A such that f ◦ g = 1 B . 8. Suppose that f : A-→ B and g : B-→ C are both bijections. Prove that g ◦ f : A-→ C is also a bijection. 9. Suppose that A has at most two elements, and f,g : A-→ A are bijections. Prove that f ◦ g = g ◦ f . 10. Suppose that A has 3 elements. Construct bijections f,g : A-→ A such that f ◦ g = g ◦ f . 11. Prove that if A and B are both countable, then so is A × B . 12. Prove that for any set A , there is no bijection between A and P ( A ). Also prove that there is always an one to one function from A to P ( A ). 2...
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## This note was uploaded on 01/23/2011 for the course MAS 111 taught by Professor Drchansongheng during the Spring '10 term at Nanyang Technological University.

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MAS111_09_assignment_13_1 - that g ◦ f = 1 A . Prove that...

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