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Discrete Mathematics Relations

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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.

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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 )}

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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

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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!

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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?

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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

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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)} 12341234 12341234 R1234 1XXXX 2XX 3X 4X

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^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

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

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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 ><=

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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.

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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 =><

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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

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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

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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

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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…

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

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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

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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

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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”

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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

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