Presented By Andrew F Conn Lecture 22 Relations and Representations November 28 th 2016 Binary relations establish a relationship between elements of two sets Definition Let and be two sets A ID: 720822
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Slide1
Discrete Structures for Computer Science
Presented By: Andrew F. Conn
Lecture #22: Relations and Representations
November 28
th
, 2016Slide2
Binary relations establish a relationship between elements of two sets
Definition:
Let
and be two sets. A binary relation from to is a subset of .In other words, a binary relation is a set of ordered pairs where and .Notation: We say that
Slide3
Example: Course Enrollments
Let
’
s say that Alice and Bob are taking CS 441. Alice is also taking Math 336. Furthermore, Charlie is taking Art 212 and Business 444. Define a relation R that represents the relationship between people and classes.Solution:Let the set denote people, so Let the set denote classes, so By definition From the above statement, we know that
So,
Slide4
A relation can also be represented as a graph
Alice
Bob
Charlie
Art 212
Business 444
CS 441
Math 336
Elements of
(i.e., people)
Elements of
(i.e., classes)
Let
’
s say that Alice and Bob are taking CS 441. Alice is also taking Math 336. Furthermore, Charlie is taking Art 212 and Business 444. Define a relation R that represents the relationship between people and classes.
Slide5
A relation can also be represented as a table
R
Art 212
Business 444CS 441Math 336Alice
X
X
Bob
X
Charlie
X
X
Let
’
s say that Alice and Bob are taking CS 441. Alice is also taking Math 336. Furthermore, Charlie is taking Art 212 and Business 444. Define a relation R that represents the relationship between people and classes.
Elements of
(i.e., courses)
Elements of
(i.e., people)
Name of the relation
Slide6
Wait, doesn
’
t this mean that relations are the same as functions?
Not quite… Recall the following definition from Lecture #9.Definition: Let and be nonempty sets. A function, , is an assignment of exactly one element of set to each element of set .Reconciling this with our definition of a relation, we see thatEvery function is also a relationNot every relation is a functionLet’s see some quick examples…
This would mean that, e.g., a person only be enrolled in one course!Slide7
Short and sweet…
Consider
Clearly a function
Can also be represented as the relationConsider the set Clearly a relationCannot be represented as a function!
Anna ●
Brian ●
Christine ●
● A
● B
● C
● D
● F
A ●
● 1
● 2Slide8
We can also define binary relations on a single set
Definition:
A
relation on the set is a relation from to . That is, a relation on the set is a subset of .Example: Let be the set . Which ordered pairs are in the relation ?Solution:1 divides everything2 divides itself and 43 divides itself4 divides itselfSo, R = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)} (1,1), (1,2), (1,3), (1,4)(2,2), (2,4)(3,3)(4,4)Slide9
Representing the last example as a graph…
Example:
Let
be the set . Which ordered pairs are in the relation ? 1
2
3
4
1
2
3
4
1
2
3
4Slide10
Tell me what you know…
Question:
Which of the following relations contain each of the pairs
and ?
Answer:
Yes
Yes
No
No
Yes
No
No
Yes
Yes
No
Yes
No
No
Yes
Yes
Yes
No
No
No
Yes
No
No
Yes
No
No
Yes
Yes
Yes
Yes
No
Yes
Yes
No
No
Yes
No
No
Yes
Yes
No
Yes
No
No
Yes
Yes
Yes
No
No
No
Yes
No
No
Yes
No
No
Yes
Yes
Yes
Yes
No
These are all relations on an infinite set!Slide11
Properties of Relations
Definition:
A relation
on a set is reflexive if for every .Note: Our “divides” relation on the set is reflexive. 1
2
3
4
Every
divides itself!
1
2
3
4
1
X
X
X
X
2
X
X
3
X
4
XSlide12
Properties of Relations
Definition:
A relation
on a set is symmetric if whenever for every . If is a relation in which and implies that , we say that is antisymmetric.Mathematically:Symmetric:
Antisymmetric:
Examples:
Symmetric:
Antisymmetric:
Slide13
Symmetric and Antisymmetric Relations
Symmetric relation
Diagonal axis of symmetry
Not all elements on the axis of symmetry need to be included in the relationAntisymmetric relationNo axis of symmetryOnly symmetry occurs on diagonalNot all elements on the diagonal need to be included in the relation1234
1
X
XXX2
X
3
X
4
X
1
2
3
4
1
X
X
X
X
2
X
X
3
X
X
4
X
X
Slide14
Properties of Relations
Definition:
A relation
on a set is transitive if whenver and , then for every .Note: Our “divides” relation on the set is transitive.
1
2
3
4
1 divides 2
2 divides 4
1 divides 4
This isn
’
t terribly interesting, but it is transitive nonetheless….
More common transitive relations include equality and comparison operators like <, >, ≤, and ≥.Slide15
Examples, redux
Question:
Which of the following relations are reflexive, symmetric, antisymmetric, and/or transitive?
Answer:
Reflexive
Symmetric
Antisymmetric
Transitive
R
1
Yes
No
Yes
Yes
R
2
No
No
Yes
Yes
R
3
Yes
Yes
No
Yes
R
4
Yes
Yes
Yes
Yes
R
5
No
No
Yes
No
R
6
No
Yes
No
NoSlide16
Relations can be combined using set operations
Example:
Let
be the relation that pairs students with courses that they have taken. Let be the relation that pairs students with courses that they need to graduate. What do the relations , , and represent?Solution: All pairs wherestudent a has taken course b ORstudent a needs to take course b to graduate All pairs whereStudent a has taken course b ANDStudent a needs course b to graduate All pairs whereStudent a needs to take course b to graduate BUTStudent a has not yet taken course b
Slide17
Relations can be combined using functional composition
Definition:
Let R be a relation from the set A to the set B, and S be a relation from the set B to the set C. The
composite of R and S is the relation of ordered pairs (a, c), where a ∈ A and c ∈ C for which there exists an element b ∈ B such that (a, b) ∈ R and (b, c) ∈ S. We denote the composite of R and S by R º S.Example: What is the composite relation of R and S?So:
Slide18
Group Work!
Problem 1:
List the ordered pairs of the relation
from to where .Problem 2: Draw the graph and table representations of the above relation.Problem 3: Is the above relation reflexive, symmetric, antisymmetric, and/or transitive? Slide19
Final Thoughts
Relations allow us to represent and reason about the relationships between sets
Relations are more general than functions
Relations are use all over…Mathematical operatorsBindings between sets of objectsEtc.Next time: n-ary relations