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Relations
Discrete Structures (CS 173)Madhusudan Parthasarathy, University of Illinois
Van Gogh
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Midterm 1
• Oct 1 in class• Skills list on website, under exams• Practice midterm and practice problems will
also be up soon.
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Last Class: Sets
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mother father
sister
grandfather
Beethoven
memy friend
Madonna
A set is an unordered collection of objects
Last Class: Sets
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mother father
sister
grandfather
Beethoven
me
Family
my friend
Madonna
Today’s class: Relations
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Beethoven
my friend
Madonna
mother father
sister
grandfather
me
parent
parent
sibling
Today’s class
• How to represent relations
• Properties and types of relations: reflexive, symmetric, transitive, partial order, etc.
• Practice proofs with relations
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Representing relationsA relation on a set is a set of ordered pairs of elements from i.e.
• Consider relation to stand for “parent” on the set of people– mother me– = {(grandfather, mother), (mother, me), (father, me), (mother, sister), (father, sister)}
• Relation stands for “sibling”– {(sister, me), (me, sister)}
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mother father
sister
grandfather
me
parent
parent
sibling
Relations with numbers
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1 2 3 4 5 6 7 8
“less than”
“divides”
“congruent mod 3”
1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8
Application: Relational Databases, SQL- A database can be seen as a relation (or sets of relations)- Represented as tables
- Query languages (like SQL)- SELECT statements combine relations to get new relations- SELECT * FROM Book WHERE price > 100.00 ORDER BY
title;- Uses JOINs, etc. Uses Boolean connectives, etc.- http://en.wikipedia.org/wiki/SQL
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Reflexivity
Reflexive: all elements relate to self
Irreflexive: no elements relate to self ( means “”)
Is irreflexive the negation of reflexive? No!
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Symmetry
Symmetric:
Antisymmetric: equivalent:
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Transitivity
Transitive:
Example of a relation that is transitive?
Not transitive?
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Practice identifying relation properties
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Reflexiv
e
Irrefle
xive
Symmetric
Antisym
metric
Transit
iveAll to self None to self
If one way then both
Never both ways if not same
if x->y->z, x->z
Practice identifying relation properties
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Reflexiv
e
Irrefle
xive
Symmetric
Antisym
metric
Transit
iveAll to self None to self
If one way then both
Never both ways if not same
if x->y->z, x->z
“less than”
“divides”
“congruent mod k”
“is square of”
Disproof of transitive
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Claim: “is square of” is not transitive.Definition: Relation on set is transitive iff
Proof of antisymmetric
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Claim: “is square of” is antisymmetric.Definition: Relation on set is antisymmetric if , or equivalently
Types of relationsPartial order:reflexive, antisymmetric, transitive
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Types of relationsLinear order: partial order (reflexive, antisymmetric, transitive) in which every pair of elements is comparable:
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Types of relationsStrict partial order: irreflexive, antisymmetric, transitive
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Types of relationsEquivalence relation: reflexive, symmetric, transitive
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Equivalence exampleRelation on : iff
contains all points on the unit circle
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Proof of equivalence
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Claim: “congruent mod k” is an equivalence relationDefinition: An equivalence relation is reflexive, symmetric, and transitive
The subset relation
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What kind of ordering is the subset () relation sets?
Types of relations
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Reflexiv
e
Irrefle
xive
Symmetric
Antisym
metric
Transit
ive
Partial Order
Linear Order
Strict Partial Order
Equivalence Relation
Things to remember
• How to illustrate a relation graphically
• Be able to identify basic properties of relations: reflexivity, symmetry, transitivity
• Types of relations: partial order, strict partial order, linear order
• When proving (or disproving) a property of a relation, write down definition of relation and property
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See you next week!
• Functions and more functions
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