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Lecture 4.1: Relations Basics
CS 250, Discrete Structures, Fall 2012
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren
04/21/23
Course Admin Mid-Term 2 Exam
How was it? Solution will be posted soon Should have the results no later than next
Tues HW3
Was due today Bonus problem? Solution will be posted soon Results should be ready no later than next
Tues
Roadmap Done with about 60% of the course We are left with two more homeworks We are left with one final exam
Cumulative – will cover everything Focus more on material covered after the two mid-terms
Please try to do your best in the remainder of the time; I know you have been working hard
Please submit your homeworks on time Still a lot of scope of improvement PLEASE DO ATTEND THE LECTURES
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Outline
Relation Definition and Examples Types of Relations Operations on Relations
Relations
Recall the definition of the Cartesian (Cross) Product:
The Cartesian Product of sets A and B, A x B, is the setA x B = {<x,y> : xA and yB}.
A relation is just any subset of the CP!! R AxB
Ex: A = students; B = courses. R = {(a,b) | student a is enrolled in class b}We often say: aRb if (a,b) belongs to R
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Relations vs. Functions
Recall the definition of a function:f = {<a,b> : b = f(a) , aA and bB}
Is every function a relation?
Draw venn diagram of cross products, relations, functions
Yes, a function is a special kind
of relation.
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Properties of Relations
Reflexivity:A relation R on AxA is reflexive if for all aA, (a,a) R.
Symmetry:A relation R on AxA is symmetric if
(a,b) R implies (b,a) R .
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Properties of Relations
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Properties of Relations - Techniques
How can we check for the reflexive property?
Draw a picture of the relation (called a “graph”).
Vertex for every element of AEdge for every element of RNow, what’s R?
{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}
12
34
Loops must exist on EVERY
vertex.
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Properties of Relations - Techniques
How can we check for the symmetric property?
Draw a picture of the relation (called a “graph”).
Vertex for every element of AEdge for every element of R
Now, what’s R?{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}
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34
EVERY edge must have a return edge.
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Properties of Relations - Techniques
How can we check for transitivity?
Draw a picture of the relation (called a “graph”).
Vertex for every element of AEdge for every element of RNow, what’s R?
{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}
12
34
A “short cut” must be present for EVERY path
of length 2.
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Properties of Relations - Techniques
How can we check for the anti-symmetric property?
Draw a picture of the relation (called a “graph”).Vertex for every element of AEdge for every element of R
Now, what’s R?{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}
12
34
No edge can have a
return edge.
Properties of Relations - Examples
Let R be a relation on People,R={(x,y): x and y have lived in the same
country}
NoIs R transitive?
Is it reflexive? Yes
Is it symmetric? Yes
Is it anti-symmetric?
No
1 2
3?
1
?
1 2
?
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Properties of Relations - Examples
Let R be a relation on positive integers,R={(x,y): 3|(x-y)}
YesIs R transitive?
Suppose (x,y) and (y,z) are in R.
Then we can write 3j = (x-y) and 3k = (y-z) Definition of “divides”
Can we say 3m = (x-z)?
Is (x,z) in R?
Add prev eqn to get: 3j + 3k = (x-y) + (y-z)
3(j + k) = (x-z)
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Properties of Relations - Techniques
Let R be a relation on positive integers,R={(x,y): 3|(x-y)}
YesIs R transitive?
Is it reflexive? Yes
Is (x,x) in R, for all x?
Does 3k = (x-x) for some k?Definition of
“divides”
Yes, for k=0.
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Properties of Relations - Techniques
Let R be a relation on positive integers,R={(x,y): 3|(x-y)}
YesIs R transitive?
Is it reflexive? Yes
Is it symmetric? Yes
Suppose (x,y) is in R.
Then 3j = (x-y) for some j.Definition of
“divides”
Yes, for k=-j.
Does 3k = (y-x) for some k?
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Properties of Relations - Techniques
Let R be a relation on positive integers,R={(x,y): 3|(x-y)}
YesIs R transitive?
Is it reflexive? Yes
Is it symmetric? Yes
Is it anti-symmetric?
No
Suppose (x,y) is in R.
Then 3j = (x-y) for some j.Definition of
“divides”
Yes, for k=-j.
Does 3k = (y-x) for some k?
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More than one relation
Suppose we have 2 relations, R1 and R2, and recall that relations are just sets! So we can take unions, intersections, complements, symmetric differences, etc.
There are other things we can do as well…
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More than one relationLet R be a relation from A to B (R AxB), and let S be a relation
from B to C (S BxC). The composition of R and S is the relation from A to C (SR AxC):
SR = {(a,c): bB, (a,b) R, (b,c) S}
SR = {(1,u),(1,v),(2,t),(3,t),(4,u)}
A B C
1
2
3
4
x
y
z
s
t
u
v
R S
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More than one relationLet R be a relation on A. Inductively define
R1 = RRn+1 = Rn R
R2 = R1R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1), (4,2)}
A A A
1
2
3
4
1
2
3
4
R R1
1
2
3
4
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More than one relationLet R be a relation on A. Inductively define
R1 = RRn+1 = Rn R
R3 = R2R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1),(4,2),(4,3)}
A A A
1
2
3
4
1
2
3
4
R R2
1
2
3
4 … = R4
= R5
= R6… 04/21/23
04/21/23
Today’s Reading Rosen 9.1