1. Lecture 4.1: Relations Basics
CS 250, Discrete Structures, Fall 2015
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren

2. Course Admin Mid-term 2 HW3 Due Being graded
Covers “Induction and Recursion” (Chapter 5)
Due 11am Nov 17 (Tuesday)
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3. Outline Relation Definition and Examples Types of Relations
Operations on Relations
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4. 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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5. 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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6. 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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7. Properties of Relations
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8. 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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9. 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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10. 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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11. 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

12. 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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13. 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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14. 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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15. 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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16. 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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17. 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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18. 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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19. 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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20. 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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21. Today’s Reading
Rosen 9.1
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