This preview shows pages 1–10. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MA1100 Lecture 18 Functions jective function • Injective function • Surjective function • ijection Bijection Chartrand: 9.3, 9.4 t Announcement ¡ Homework 4 due next Tuesday ¡ Homework 3 scores in Gradebook ¡ Midterm test scores in Gradebook 2 Lecture 18 j t i f t i Injective function A B g a 1 There are two elements in A that maps to a same b c d 2 3 element in B A B f negation a b 1 2 No two elements in A maps to a same element in B very pair of (different) elements in 3 c d 3 Every pair of (different) elements in A have different images in B Lecture 18 j t i f t i Injective function A B f a 1 Formulate the statement athematically b c d 2 3 mathematically Every pair of (different) elements in A have different images in B " x, y œ A [f(x) ∫ f(y)] Is this correct? " x, y œ A [If x ∫ y , then f(x) ∫ f(y)] 4 Lecture 18 You can’t give me two different injection at j t i f t i the same spot Injective function efinition Definition Let f: A Ø B be a function from set A to set B. Suppose " x, y œ A [if x ∫ y , then f(x) ∫ f(y)] he function is called an jective function A B f The function is called an injective function . We can also say: a b 1 2 f is a onetoone function f is an injection 5 c d 3 Lecture 18 r k i d f i i t i Working definition f: A is an jective function " x, y œ A [if x ∫ y , then f(x) ∫ f(y)] If f: A Ø B is an injective function : Contrapositive " x, y œ A [if f(x) = f(y) , then x = y] If f: A Ø B is not an injective function : working definition $ x, y œ A such that [x ∫ y and f(x) = f(y)] Negation 6 Lecture 18 E.g. The sumofdivisor function s is not injective. il lif pl Daily life examples Example (Daily life) Let A be the set of MA1100 students et f: A Ø : et of student numbers Let f: A B injective B: set of student numbers f maps every student to his/her student number ject e Let g: A Ø C C: set of all integers g maps every student to his/her MA1100 test score not injective Let h: A Ø D D: set of mobile phone numbers 7 may not be a function h maps every student to his/her mobile number Lecture 18 l f ti (Vi liz ) Real function (Visualize) Example f: R Ø R defined by f(x) = 5x + 3 is injective . y Visualization Plot the graph of the real function. bserve that ny horizontal line x Observe that any horizontal line intersects with the graph at exactly one point . This means every value on the yaxis corresponds to exactly one value on e x xis 8 the xaxis. So f is an injective function. Lecture 18 l f t i ( f r l r t ) " x, y œ A [if x ∫ y , then f(x) ∫ f(y)] Real function (formal argument) Example f: R Ø R defined by f(x) = 5x + 3 is injective . " x, y œ R [if f(x) = f(y) , then x = y] Proof (Working definition) f(x) = f(y) ﬂ 5x + 3 = 5y + 3 hypothesis ﬂ 5x = 5y ﬂ x = y conclusion 9 So f is an injective function....
View
Full
Document
This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.
 Spring '10
 VT

Click to edit the document details