1. Lecture 4.1: Relations Basics
CS 250, Discrete Structures, Fall 2014
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren

2. Course Admin Graded HW2 HW3 Due Will distribute today
Covers “Induction and Recursion” (Chapter 5)
Due 11am Nov 6 (Thursday)
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3. Course Admin Mid Term 2 Nov 6 (Thur)
Covers “Induction and Recursion” (Chapter 5)
Review Oct 30 (Thu)
Study topics posted
Sample practice exam will be given out soon
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4. Traveling Next Week At a conference (Nov 3-7):
Mid-term proctored by TA
No class next Tuesday (Nov 4) – prepare for exam
Office hrs extended – 3-5pm Thursday.
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5. Outline Relation Definition and Examples Types of Relations
Operations on Relations
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6. A x B = {<x,y> : xA and yB}.
Relations
Recall the definition of the Cartesian (Cross) Product:
The Cartesian Product of sets A and B, A x B, is the set
A x B = {<x,y> : xA and yB}.
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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7. Relations vs. Functions
Recall the definition of a function:
f = {<a,b> : b = f(a) , aA and bB}
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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8. Properties of Relations
Reflexivity:
A relation R on AxA is reflexive if for all aA, (a,a) R.
Symmetry:
A relation R on AxA is symmetric if
(a,b)  R implies (b,a)  R .
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9. Properties of Relations
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10. 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 A
Edge 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)} 1 2 3 4
Loops must exist on EVERY vertex.
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11. 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 A
Edge 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)}
EVERY edge must have a return edge. 1 2 3 4
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12. Properties of Relations - Techniques
How can we check for transitivity?
Draw a picture of the relation (called a “graph”).
Vertex for every element of A
Edge 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)} 1 2 3 4
A “short cut” must be present for EVERY path of length 2.
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13. 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 A
Edge 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)}
No edge can have a return edge. 1 2 3 4

14. Properties of Relations - Examples
Let R be a relation on People,
R={(x,y): x and y have lived in the same country} 1 ? 1 2 ? 1 2 3 ?
Is R transitive? No
Is it symmetric?
Yes
Is it reflexive?
Yes
Is it anti-symmetric? No
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15. Properties of Relations - Examples
Let R be a relation on positive integers,
R={(x,y): 3|(x-y)}
Suppose (x,y) and (y,z) are in R.
Definition of “divides”
Then we can write 3j = (x-y) and 3k = (y-z)
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)
Is R transitive?
Yes
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16. Properties of Relations - Techniques
Let R be a relation on positive integers,
R={(x,y): 3|(x-y)}
Is (x,x) in R, for all x?
Definition of “divides”
Does 3k = (x-x) for some k?
Yes, for k=0.
Is R transitive?
Yes
Is it reflexive?
Yes
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17. Properties of Relations - Techniques
Let R be a relation on positive integers,
R={(x,y): 3|(x-y)}
Suppose (x,y) is in R.
Definition of “divides”
Then 3j = (x-y) for some j.
Does 3k = (y-x) for some k?
Yes, for k=-j.
Is R transitive?
Yes
Is it symmetric?
Yes
Is it reflexive?
Yes
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18. Properties of Relations - Techniques
Let R be a relation on positive integers,
R={(x,y): 3|(x-y)}
Suppose (x,y) is in R.
Definition of “divides”
Then 3j = (x-y) for some j.
Does 3k = (y-x) for some k?
Yes, for k=-j.
Is R transitive?
Yes
Is it symmetric?
Yes
Is it reflexive?
Yes
Is it anti-symmetric? No
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19. 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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20. More than one relation SR = {(a,c):  bB, (a,b)  R, (b,c)  S}
Let 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 (SR  AxC):
SR = {(a,c):  bB, (a,b)  R, (b,c)  S} A B C 1 2 3 4 x y z s t u v R S
SR = {(1,u),(1,v),(2,t),(3,t),(4,u)}
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21. R2 = R1R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1), (4,2)}
More than one relation
Let R be a relation on A. Inductively define
R1 = R
Rn+1 = Rn  R A A A R R1 1 1 1 2 2 2 3 3 3 4 4 4
R2 = R1R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1), (4,2)}
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22. R3 = R2R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1),(4,2),(4,3)}
More than one relation
Let R be a relation on A. Inductively define
R1 = R
Rn+1 = Rn  R A A A R R2 1 1 1 2 2 2 3 3 3
… = R4
= R5
= R6… 4 4 4
R3 = R2R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1),(4,2),(4,3)}
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23. Today’s Reading
Rosen 9.1
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