Lecture_17_complete - MA1100 Lecture 17 Functions Injection...

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1 MA1100 Lecture 17 Functions Injection Surjection Bijection
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MA1100 Lecture 17 2 Announcement ± Homework 4 due next Tuesday ± Homework 3 scores in Gradebook ± Mid-term test scores will be in Gradebook by today. ± Students who missed tutorial class this week can pick up their test scripts from me.
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MA1100 Lecture 17 3 Lesson Plan (week 9 to 13) HW4 Tut 7 Functions 19. Number Th. 18. Functions Week 10 Tut 6 Relations 17. Functions 16. Functions Week 9 Tut 10 Misc Tut 9 Number Th. Tut 8 Functions Tutorial HW5 23. Revision lecture 22. Revision lecture Week 12 Week 13 21. Number Th. 20. Number Th. Week 11 HW Fri Lecture Tue Lecture
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MA1100 Lecture 17 4 Injection a b c 1 2 3 A B g a b c 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 Every pair of (different) elements in A have different images in B
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MA1100 Lecture 17 5 Injection a b c 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)] " x, y œ A[ f ( x ) f(y)]
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MA1100 Lecture 17 6 Injection a b c 1 2 3 A B f " 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 The function is called an injection. We can also say: f is a one-to-one function f is an injective function
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MA1100 Lecture 17 7 Injection " x, y œ A [if x y , then f(x) f(y)] If f: A Ø B is an injection: Contrapositive " x, y œ A [if f(x) = f(y) , then x = y] If f: A Ø B is not an injection: $ x, y œ A such that [x y and f(x) = f(y)] Negation working definition
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MA1100 Lecture 17 8 Injection Let A be the set of MA1100 students Let f: A Ø B Example (Daily life) injection B: set of NUS student numbers f maps every student to his/her student number Let g: A Ø C not injection C: set of all integers g maps every student to his/her MA1100 test score Let h: A Ø D may not be a function D: set of mobile phone numbers h maps every student to his/her mobile number
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MA1100 Lecture 17 9 Injection f: R Ø R defined by f(x) = 5x + 3 is an injection. 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 injection.
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MA1100 Lecture 17 10 Injection f: R Ø R defined by f(x) = 5x + 3 is an injection. " x, y œ R [if f(x) = f(y) , then x = y] Example Proof " x, y œ A [if x y , then f(x) f(y)] Contrapositive proof f(x) = f(y) 5x + 3 = 5y + 3 5x = 5y x = y hypothesis conclusion So f is an injection
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MA1100 Lecture 17 11 Injection f: R Ø R defined by f(x) = x 2 + 1 is not an injection.
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This note was uploaded on 03/19/2012 for the course SCIENCE MA1100 taught by Professor Forgot during the Fall '08 term at National University of Singapore.

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Lecture_17_complete - MA1100 Lecture 17 Functions Injection...

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