EDAA40
Discrete Structures in Computer Science
Seminar 2
A recap: sets, relations, functions
Jörn W. Janneck, Dept. of Computer Science, Lund University
sets
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how to specify them, incl. recursive definition
elementhood, extension, subsets
operations, set algebra, generalized union/intersection
extensionality, cardinality, how to compare cardinals
power sets
2
specifying sets
enumeration of its elements
set builder notation / set comprehensions
flavor 1
flavor 2
bad flavor
3
operations on sets
A
B
A
B
A
B
union
all elements that are in A or B or both
intersection
all elements that are both in A and B
difference
all elements that are in A and not in B
4
generalized union & intersection
Let S be a set of sets.
Often, S is a family of sets. Then we write...
Note:
Here be dragons!
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extensionality, cardinality
A set is defined by the elements it contains (its extension).
The number of elements in a
set A is called its cardinality.
alternative syntax
For any two sets A and B, if there is an injection
then
For any two sets A and B, if there is a bijection
then
A and B are equinumerous. We write
Cantor-Schröder-Bernstein (CSB) theorem:
If there are two injections
then there exists a bijection
and
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power sets
The power set of a set A is
the set of all its subsets.
alternative syntax
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relations
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ordered pairs, tuples, Cartesian products, relations
source, target, domain, range
operations on relations
properties of relations
equivalence relations
order relations
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ordered pairs, tuples, relations
ordered pair
n-tuple
The (cartesian) product of a pair of sets, or more generally a finite
family of sets, is the set of all ordered pairs or n-tuples.
An n-place relation R over A1 x … x An is a subset of that product:
Special case: (binary, dyadic) relation R from A to B :
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source, target, domain, range
actual values, left
domain
actual values, right
range
A
source
B
target
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operations on relations
For a binary relation
its converse (inverse) is the relation
For a binary relation
its complement is the relation
Given two binary relations
and
their composition is a binary relation on
Given a binary relation
from A to B, for any
its image under R, written R(a), is defined as
Can be “lifted” to subsets
:
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properties
A binary relation
is reflexive iff for all
A binary relation
such that
is irreflexive iff there is no
A binary relation
is transitive iff for all
A binary relation
is symmetric iff for all
A binary relation
is asymmetric iff for all
A binary relation
is antisymmetric iff for all
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equivalence relations
A binary relation
1. reflexive
is an equivalence relation iff it is
2. symmetric
3. transitive
Given a set A, a partition of A is a set of pairwise disjoint sets
, such that
Given a set A and an equivalence relation
define the equivalence class of a
as
on A, for any
we
Given a set A and an equivalence relation
is defined as
on A, the quotient (set)
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order relation, poset
A binary relation
order iff it is
1. reflexive
is an ( inclusive or non-strict) (partial)
2. antisymmetric
A pair
where A is a set and
called a partially ordered set or poset.
A binary relation
1. irreflexive
3. transitive
a partial order on A is
is a strict (partial) order iff it is
2. transitive
A binary relation
is a ( non-strict) total (or linear) order
iff it is
1. reflexive
2. antisymmetric
3. transitive
4. total (complete):
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transitive closure
The transitive closure
defined as follows:
of a binary relation
is
Note: This is a union of infinitely many sets, so: dragon warning!
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functions
• functions as special relations
• operations on functions
• properties of functions
– injections, surjections, bijections
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functions, terminology
is a function iff
A relation
We then also write
actual values, left
domain
domain
actual values, right
range
range
A
source
domain
B
target
codomain
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operations on/with functions
, its restriction to a set
Given a function
is defined as
alternative syntax
Given a function
and a set
the image of X under f is defined as
Given an endofunction
the closure of X under f
such that
Given functions
and a set
,
is defined as the smallest
and
defined as:
,
their composition
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injection, surjection, bijection
is injective (and thus an injection) iff
A function
Notation:
is surjective (and thus an surjection) iff
A function
Notation:
A function
is bijective (and thus an bijection) iff
it is both injective and surjective.
Notation:
A
a
b
c
B
1
2
3
4
injection, one-to-one
A
a
b
c
d
B
1
2
3
surjection, onto
A
a
b
c
d
B
1
2
3
4
bijection
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infinity
• infinite sets
• countable sets, transfinite cardinals
• comparing the number of elements in infinite sets
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infinite sets
A is infinite if it is equinumerous to a proper subset of itself.
That is, there is some S such that
A is denumerable (countable) if it is equinumerous to the
natural numbers, i.e.
The cardinality of the natural numbers
(and thus all denumerable sets) has a name:
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