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Lecture 1b
Review of Discrete Structures +
The Pigeonhole Principle
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Recall some Discrete Structures
Relation
Homogeneous relation (relation on)
Function, partial function
Injection, surjection, bijection
Injection = one-to-one function
Surjection = onto function
Bijection = one-to-one correspondence
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Recall some Discrete Structures
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a b f(a) f(b) contrapositive
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A surjective function is a function whose image is equal to its
codomain. Equivalently, a function f with domain X and codomain Y
is surjective if for every y in Y there exists at least one x in X
with .
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The term surjective and the related terms injective and
bijective were introduced by Nicolas Bourbaki, the pseudonym for a
group of mainly French 20th-century mathematicians who wrote a
series of books presenting an exposition of modern advanced
mathematics, beginning in 1935.
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Partial Functions
f: X Y is a partial function iff
xX: |f(x)| 1
Sometimes also denoted f: X Y using an arrow with vertical
stroke.
Another definition
A partial function f: X Y is a function f: X' Y, where X' X.
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Partial functions
It generalizes the concept of a function f: X Y by not forcing f
to map every element of X to an element of Y.
If X' = X, then f is called a total function and is equivalent
to a function.
Partial functions are often used when the exact domain, X' , is
not known (e.g. many functions in computability theory).
Specifically, we will say that for any x X, either: f(x) = y Y
(it is defined as a single element in Y) or
f(x) is undefined.
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Partial functions
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For example we can consider the square root function restricted
to the integers Thus g(n) is only defined for n that are perfect
squares (i.e. 0, 1, 4, 9, 16, ...). So, g(25) = 5, but g(26) is
undefined.
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Partial functions
There are two distinct meanings in current mathematical usage
for the notion of the domain of a partial function.
Most mathematicians, including recursion theorists, use the term
"domain of f" for the set of all values x such that f(x) is defined
(X' above).
But some, particularly category theorists, consider the domain
of a partial function f:XY to be X, and refer to X' as the domain
of definition.
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Partial functions
Similarly, the term range can refer to either the codomain Y or
the image f(X) = {yY: xX: y=f(x)} of a function. In other words,
the range of f is the set of all values that the function assigns
to its arguments.
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Partial functions
A partial function is said to be injective or surjective when
the total function given by the restriction of the partial function
to its domain of definition is.
A partial function may be both injective and surjective, but the
term bijection generally only applies to total functions.
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A useful rule often used without thinking
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The Pigeonhole Principle
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Colloquial version: If n pigeons fly into m < n pigeonholes,
then at least one pigeonhole will have more than one pigeons
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The Pigeonhole Principle
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Illustration
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Illustration
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10 pigeons 9 holes
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Pigeonhole principle
In more formal terms: if A and B are finite sets and |A| >
|B| then no function A B is an injection.
This is not true for infinite sets!
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Can be proved formally, e.g., Epp 4th ed. 561, Gopalakrishnan
85
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Examples of usage
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Examples of usage
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Length n means n arrows which is n+1 nodes
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Resources
Hein 111
Epp 4t ed. 554
Gopalakrishnan 85
Sipser 3rd 78
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