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lecture18&iuml;&frac14;ˆcomplete&iuml;&frac14;‰

# lecture18&iuml;&frac14;ˆcomplete&iuml;&frac...

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MA1100 Lecture 18 Functions Injective function Surjective function Bijection Chartrand: 9.3, 9.4

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A t Announcement Homework 4 due next Tuesday Homework 3 scores in Gradebook Mid-term test scores in Gradebook 2 Lecture 18
I j ti f ti Injective function A B g a b 1 2 There are two elements in A that maps to a same l t i B c d 3 element in B ti A B f negation a b 1 2 3 No two elements in A maps to a same element in B Every pair of (different) elements in 3 c d A have different images in B Lecture 18

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I j ti f ti Injective function A B f a b 1 2 Formulate the statement mathematically c d 3 E i f (diff t) l t i A h diff t i i B 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 I j ti f ti the same spot Injective function 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)] The function is called an injective function A B f . We can also say: a b c 1 2 3 f is a one-to-one function f is an injection 5 d Lecture 18

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W rki d fi iti Working definition If f: A Ø B is an injective function : " x, y œ A [if x y , then f(x) f(y)] C t iti Contrapositive " x, y œ A [if f(x) = f(y) , then x = y] ki d fi iti If f: A Ø B is not an injective function : Negation working definition \$ x, y œ A such that [x y and f(x) = f(y)] 6 Lecture 18 E.g. The sum-of-divisor function s is not injective.
D il lif pl Daily life examples Example (Daily life) Let A be the set of MA1100 students Let f: A Ø B B: set of student numbers injective 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 not injective Let h: A Ø D D: set of mobile phone numbers may not be a function h maps every student to his/her mobile number Lecture 18

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R l f ti (Vi liz ) Real function (Visualize) Example f: R Ø R defined by f(x) = 5x + 3 is injective . y Vi li ti Visualization Plot the graph of the real function. Observe that any horizontal line x 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 8 the x-axis. So f is an injective function. Lecture 18
R l f ti (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 . Proof (Working definition) " x, y œ R [if f(x) = f(y) , then x = y] f(x) = f(y) 5x + 3 = 5y + 3 hypothesis 5x = 5y x = y conclusion 9 So f is an injective function. Lecture 18

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R l f ti ( i liz ) Real function (visualize) Example f: R Ø R defined by f(x) = x 2 + 1 is not injective . y Visualization Plot the graph of the real function.
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lecture18&iuml;&frac14;ˆcomplete&iuml;&frac...

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