Functions MATHEMATICS

SIR SYED UNIVERSITY OF ENGINEERING & TECHNOLOGY

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GROUP MEMBERS

• Burhanuddin Shabbir (2015-CS-101)

• Muzzamil Jaffri (2015-CS-105)• Umair Abdul Rahman (2015-CS-77)• Hasnain Ahmed (2015-CS-300)• Hussain Ansari (2015-CS-78)

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Definition of Functions

• Given any sets A, B, a function f from (or “mapping”) A to B (f:AB) is an assignment of exactly one element f(x)B

to each element xA.

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Graphical Representations

• Functions can be represented graphically in several ways:

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A B

a b

f

f

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• x

y

PlotGraphLike Venn diagrams

A B

Some Function Terminology

• If f:AB, and f(a)=b (where aA & bB), then:• A is the domain of f. • B is the codomain of f.• b is the image of a under f.• a is a pre-image of b under f.• In general, b may have more than one pre-

image.• The range RB of f is {b | a f(a)=b }.

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Range vs. Codomain• Suppose that: “f is a function mapping students in

this class to the set of grades {A,B,C,D,E}.”

• At this point, you know f’s codomain is: __________, and its range is ________.

• Suppose the grades turn out all As and Bs.

• Then the range of f is _________, but its codomain is __________________.

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{A,B,C,D,E} unknown!

{A,B}still {A,B,C,D,E}!

One-to-One Functions• A function is one-to-one (1-1), or injective, or an injection,

iff every element of its range has only one pre-image. • Only one element of the domain is mapped to any given

one element of the range.• Domain & range have same cardinality. What about

codomain?• Formally: given f:AB

“x is injective” : ( x1,x2 X,if F(x1)=F(x2)then x1=x2)7

One-to-One Illustration• Graph representations of functions that are (or

not) one-to-one:

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One-to-one

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•Not one-to-one

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•Not even a function!

Onto (Surjective) Functions

• A function f:AB is onto or surjective or a surjection iff its range is equal to its codomain (bB, aA: f(a)=b).

• An onto function maps the set A onto (over, covering) the entirety of the set B, not just over a piece of it.

• e.g., for domain & codomain R, x3 is onto, whereas x2

isn’t. (Why not?)

• Formally: given f:AB

“x is surjective” : (yY xX such that F(x)=y)9

Illustration of Onto

• Some functions that are or are not onto their codomains:

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Onto(but not 1-1)

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•Not Onto(or 1-1)

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Both 1-1and onto

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•1-1 butnot onto

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ARROW DIAGRAM OF ONTO & ONE TO ONE

FUNCTION

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Bijections

• A function f is a one-to-one correspondence, or a bijection, or reversible, or invertible, iff it is both one-to-one and onto.

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THE PIGEONHOLE PRINCIPLE

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• In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. This theorem is exemplified in real life by truisms like "there must be at least two left gloves or two right gloves in a group of three gloves".

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EXAMPLE OF PIGEONHOLE

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