Presentation is loading. Please wait.

Presentation is loading. Please wait.

Discrete Mathematics Relations. What is a relation ^Relation generalizes the notion of functions. ^Recall: A function takes EACH element from a set and.

Similar presentations


Presentation on theme: "Discrete Mathematics Relations. What is a relation ^Relation generalizes the notion of functions. ^Recall: A function takes EACH element from a set and."— Presentation transcript:

1 Discrete Mathematics Relations

2 What is a relation ^Relation generalizes the notion of functions. ^Recall: A function takes EACH element from a set and maps it to a UNIQUE element in another set  f: X  Y   x  X,  y such that f(x) = y ^Let A and B be sets. A binary relation R from A to B is a subset of A  B  Recall: A x B = {(a, b) | a  A, b  B}  aRb: (a, b)  R. ^Application  Relational database model is based on the concept of relation.

3 What is a relation ^Example  Let A be the students in a the CS major  A = {A yşe, B arış, C anan, Da vut }  Let B be the courses the department offers  B = { BİM111, BİM122, BİM124 }  We specify relation R  A  B as the set that lists all students a  A enrolled in class b  B  R = {(A yşe, BİM111 ), (B arış, BİM122 ), (B arış, BİM124 ), (Da vut, BİM122 ), (Da vut, BİM124 )}

4 More relation examples ^Another relation example:  Let A be the cities in Turkey  Let B be the districts in Turkey  We define R to mean a is a district in city b  Thus, the following are in our relation:  ( Bakırköy, İstanbul)  ( Keçiören, Ankara )  ( Nilüfer, Bursa )  ( Tepebaşı, Eskişehir )  etc… ^Most relations we will see deal with ordered pairs of integers

5 Representing relations BİM111 BİM122 BİM124 Ayşe Barış Canan Davut BİM111BİM122BİM124 A yşe X B arış XX C anan Da vut XX We can represent relations graphically: We can represent relations in a table: Not valid functions!

6 Relations vs. functions  If R  X  Y is a relation, then is R a function?  If f: X  Y is a function, then is f a relation?

7 Relations on a set ^A relation on the set A is a relation from A to A  In other words, the domain and co-domain are the same set  We will generally be studying relations of this type

8 Relations on a set ^Let A be the set { 1, 2, 3, 4 } ^Which ordered pairs are in the relation R = { (a,b) | a divides b } ^R = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)} R1234 1XXXX 2XX 3X 4X

9 ^Consider some relations on the set Z ^Are the following ordered pairs in the relation? ^ (1,1) (1,2) (2,1) (1,-1) (2,2) ^R 1 = { (a,b) | a≤b } ^R 2 = { (a,b) | a>b } ^R 3 = { (a,b) | a=|b| } ^R 4 = { (a,b) | a=b } ^R 5 = { (a,b) | a=b+1 } ^R 6 = { (a,b) | a+b≤3 } More examples X X X X X X X X X X X X X X X

10 Relation properties ^Six properties of relations we will study:  Reflexive  Irreflexive  Symmetric  Asymmetric  Antisymmetric  Transitive

11 Reflexivity vs. Irreflexivity ^Reflexivity  Definition: A relation is reflexive if  (a,a)  R for all a  A ^Irreflexivity  Definition: A relation is irreflexive if  (a,a)  R for all a  A ^Examples  Is the “divides” relation on Z + reflexive?  Is the “” (not ) relation on a P(A) irreflexive? xxoox irreflexive ooxxoreflexive  ><=

12 Reflexivity vs. Irreflexivity ^A relation can be neither reflexive nor irreflexive ^Example?  A = {1, 2}, R = {(1, 1)}  It is not reflexive, since (2, 2)  R,  It is not irreflexive, since (1, 1)  R.

13 Symmetry, Asymmetry, Antisymmetry ^A relation is symmetric if  for all a, b  A, (a,b)  R (b,a)  R ^A relation is asymmetric if  for all a, b  A, (a,b)  R  (b,a)  R ^A relation is antisymmetric if  for all a, b  A, ((a,b)  R  (b,a)  R)  a=b  (Second definition) for all a, b  A, ((a,b)  R  a  b)  (b,a)  R) xoooooantisymmetric xxxxooasymmetric oxxoxxsymmetric isTwinOf  =><

14 Notes on *symmetric relations ^A relation can be neither symmetric or asymmetric  R = { (a,b) | a=|b| }  This is not symmetric  -4 is not related to itself  This is not asymmetric  4 is related to itself  Note that it is antisymmetric

15 Transitivity ^A relation is transitive if  for all a, b, c  A, ((a,b)R  (b,c)R)  (a,c)R ^If a < b and b < c, then a < c  Thus, < is transitive ^If a = b and b = c, then a = c  Thus, = is transitive

16 Transitivity examples ^Consider isAncestorOf()  Let A yşe be B arış ’s ancestor, and B arış be C anan ’s ancestor  Thus, A yşe is an ancestor of B arış, and B arış is an ancestor of C anan  Thus, A yşe is an ancestor of C anan  Thus, isAncestorOf() is a transitive relation ^Consider isParentOf()  Let A yşe be B arış ’s parent, and B arış be C anan ’s parent  Thus, A yşe is a parent of B arış, and B arış is a parent of C anan  However, A yşe is not a parent of C anan  Thus, isParentOf() is not a transitive relation

17 Summary of properties of relations reflexive a (a, a)  R irreflexive a (a, a)  R symmetric   a, b  A, (a,b)  R  (b,a)  R asymmetric  a, b  A, (a,b)  R  (b,a)  R antisymmetric  a, b  A, ((a,b)  R  (b,a)  R)  a=b (*) for all a, b  A, ((a,b)  R  a  b)  (b,a)  R) transitive  a, b, c  A, ((a,b)  R  (b,c)  R)  (a,c)  R (*) Alternative definition…

18 Combining relations ^There are two ways to combine relations R 1 and R 2  Via Set operators  Via relation “composition”

19 Combining relations via Set operators ^Consider two relations R ≥ and R ≤  R ≥ U R ≤ = all numbers ≥ OR ≤  That’s all the numbers  R ≥ ∩ R ≤ = all numbers ≥ AND ≤  That’s all numbers equal to  R ≥  R ≤ = all numbers ≥ or ≤, but not both  That’s all numbers not equal to  R ≥ - R ≤ = all numbers ≥ that are not also ≤  That’s all numbers strictly greater than  R ≤ - R ≥ = all numbers ≤ that are not also ≥  That’s all numbers strictly less than ^Note that it’s possible the result is the empty set

20 Combining via relational composition ^Similar to function composition ^Let R be a relation from A to B, and S be a relation from B to C  Let a  A, b  B, and c  C  Let (a,b)  R, and (b,c)  S  Then the composite of R and S consists of the ordered pairs (a,c)  We denote the relation by S ◦ R  Note that S comes first when writing the composition!  (a, c)  S ◦ R if  b such that (a, b)  R, and (b,c)  S

21 Combining via relational composition ^Let M be the relation “is mother of” ^Let F be the relation “is father of” ^What is M ◦ F?  If (a,b)  F, then a is the father of b  If (b,c)  M, then b is the mother of c  Thus, M ◦ F denotes the relation “maternal grandfather” ^What is F ◦ M?  If (a,b)  M, then a is the mother of b  If (b,c)  F, then b is the father of c  Thus, F ◦ M denotes the relation “paternal grandmother” ^What is M ◦ M?  If (a,b)  M, then a is the mother of b  If (b,c)  M, then b is the mother of c  Thus, M ◦ M denotes the relation “maternal grandmother”

22 Combining via relational composition ^Given relation R  R ◦ R can be denoted by R 2  R 2 ◦ R = (R ◦ R) ◦ R = R 3  Example: M 3 is your mother’s mother’s mother


Download ppt "Discrete Mathematics Relations. What is a relation ^Relation generalizes the notion of functions. ^Recall: A function takes EACH element from a set and."

Similar presentations


Ads by Google