lecture18 - MA1100 Lecture 18 Functions Injective function...

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MA1100 Lecture 18 Functions Injective function Surjective function Bijection Chartrand: 9.3, 9.4
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2 Injective function a b c d 1 2 3 A B g a b c d 1 2 3 A B f There are two elements in A that maps to a same element in B No two elements in A maps to a same element in B negation Lecture 18
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3 Injective function a b c d 1 2 3 A B f Every pair of (different) elements in A have different images in B Formulate the statement mathematically " x, y œ A [If x y , then f(x) f(y)] Lecture 18
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4 Injective function a b c d 1 2 3 A B f The function is called an injective function . We can also say: f is a one-to-one function f is an injection " x, y œ A [if x y , then f(x) f(y)] Definition Let f: A Ø B be a function from set A to set B. Suppose Lecture 18
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5 Working definition " 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 : $ x, y œ A such that [x y and f(x) = f(y)] Negation working definition Lecture 18 E.g. The sum-of-divisor function s is not injective.
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6 Daily life examples Let A be the set of MA1100 students Let f: A Ø B Example (Daily life) B: set of student numbers f maps every student to his/her student number Let g: A Ø C C: set of all integers g maps every student to his/her MA1100 test score Let h: A Ø D D: set of mobile phone numbers h maps every student to his/her mobile number Lecture 18
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7 Real function (Visualize) f: R Ø R defined by f(x) = 5x + 3 is injective . Example x y Visualization Plot the graph of the real function. 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 the x-axis. So f is an injective function. Lecture 18
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8 Real function (formal argument) f: R Ø R defined by f(x) = 5x + 3 is injective . " x, y
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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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lecture18 - MA1100 Lecture 18 Functions Injective function...

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