1. Relations Discrete Structures (CS 173)
Van Gogh
Discrete Structures (CS 173)
Madhusudan Parthasarathy, University of Illinois

2. Midterm 1 Oct 1 in class Skills list on website, under exams
Practice midterm and practice problems will also be up soon.

3. Last Class: Sets A set is an unordered collection of objects
grandfather
Madonna
mother
father
Beethoven
sister
my friend me

4. Last Class: Sets Family grandfather Madonna mother father Beethoven
sister
my friend me

5. Today’s class: Relations
mother
father
sister
grandfather me
parent
sibling
Madonna
Beethoven
my friend

6. Today’s class How to represent relations
Properties and types of relations: reflexive, symmetric, transitive, partial order, etc.
Practice proofs with relations

7. Representing relations
A relation 𝑅 on a set 𝐴 is a set of ordered pairs of elements from 𝐴
i.e. 𝑅 ⊆𝐴 ×𝐴
Consider relation 𝑅 to stand for “parent” on the set of people
mother 𝑅 me
𝑅 = {(grandfather, mother), (mother, me), (father, me), (mother, sister), (father, sister)}
Relation 𝑆 stands for “sibling”
𝑆={(sister, me), (me, sister)}
mother
father
sister
grandfather me
parent
sibling

8. Relations with numbers
“less than” 1 2 3 4 5 6 7 8
“divides” 1 2 3 4 5 6 7 8
“congruent mod 3” 1 2 3 4 5 6 7 8

9. 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 > ORDER BY title;
Uses JOINs, etc. Uses Boolean connectives, etc.

10. Reflexivity
Reflexive: all elements relate to self 𝐹𝑜𝑟 𝑎𝑙𝑙 𝑥∈𝐴, 𝑥𝑅𝑥 Irreflexive: no elements relate to self 𝐹𝑜𝑟 𝑎𝑙𝑙 𝑥∈𝐴, 𝑥𝑅𝑥 (𝑅 means “𝑅 does not hold”) Is irreflexive the negation of reflexive? No!

11. Symmetry
Symmetric: ∀𝑥, 𝑦∈𝐴, 𝑥𝑅𝑦→𝑦𝑅𝑥 Antisymmetric: ∀𝑥, 𝑦∈𝐴 with 𝑥≠𝑦, 𝑥𝑅𝑦→𝑦𝑅𝑥 equivalent: ∀𝑥, 𝑦∈𝐴, 𝑥𝑅𝑦∧ 𝑦𝑅𝑥→ 𝑥=𝑦

12. Transitivity
Transitive: ∀𝑥,𝑦,𝑧∈𝐴, 𝑥𝑅𝑦∧𝑦𝑅𝑧→𝑥𝑅𝑧 Example of a relation that is transitive? Not transitive?

13. Practice identifying relation properties
Never both ways if not same
If one way then both
All to self
None to self
if x->y->z, x->z
Antisymmetric
Reflexive
Irreflexive
Symmetric
Transitive

14. Practice identifying relation properties
Never both ways if not same
If one way then both
All to self
None to self
if x->y->z, x->z
Antisymmetric
Symmetric
Reflexive
Irreflexive
Transitive
“less than”
“divides”
“congruent mod k”
“is square of”

15. Disproof of transitive
Claim: “is square of” is not transitive.
Definition: Relation 𝑅 on set 𝐴 is transitive iff ∀𝑥,𝑦,𝑧∈𝐴, 𝑥𝑅𝑦∧𝑦𝑅𝑧→𝑥𝑅𝑧
Emphasize writing down definition

16. Proof of antisymmetric
Claim: “is square of” is antisymmetric.
Definition: Relation 𝑅 on set 𝐴 is antisymmetric if ∀𝑥, 𝑦∈𝐴 with 𝑥≠𝑦, 𝑥𝑅𝑦→𝑦𝑅𝑥, or equivalently ∀𝑥, 𝑦∈𝐴, 𝑥𝑅𝑦∧ 𝑦𝑅𝑥→ 𝑥=𝑦
Note that second case is often more useful

17. Types of relations
Partial order: reflexive, antisymmetric, transitive

18. Types of relations
Linear order: partial order (reflexive, antisymmetric, transitive) in which every pair of elements is comparable: ∀𝑥,𝑦∈𝐴, (𝑥𝑅𝑦 or 𝑦𝑅𝑥)

19. Types of relations
Strict partial order: irreflexive, antisymmetric, transitive

20. Types of relations
Equivalence relation: reflexive, symmetric, transitive

21. Equivalence example
Relation 𝐶 on 𝑅 2 : 𝑥,𝑦 𝐶(𝑎, 𝑏) iff 𝑥 2 + 𝑦 2 = 𝑎 2 + 𝑏 2
0,1 𝐶 contains all points on the unit circle

22. Proof of equivalence
Claim: “congruent mod k” is an equivalence relation
Definition: An equivalence relation is reflexive, symmetric, and transitive

23. The subset relation
What kind of ordering is the subset () relation sets?

24. Types of relations Antisymmetric Symmetric Irreflexive Reflexive
Transitive
Partial Order
Strict Partial Order
Linear Order
Equivalence Relation

25. 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

26. See you next week!
Functions and more functions