lecture18(complete)

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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 ¡ Mid-term 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 one-to-one 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 sum-of-divisor 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 y-axis corresponds to exactly one value on e x xis 8 the x-axis. 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....
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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.

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