From names to ID numbers. From zip codes to cities. From countries to leaders.
From years to events. CS 2233 Discrete Mathematical Structures. Relations – 3.
Relations
2
Relations
Relations
A relation on a set A is a subset of A × A. A relation from A to B is a subset of A × B.
In Greek, the mappings between orthography and phonology are highly consistent, whereas in English the relationship between letters and sounds is much more variable. (Maria Ktori, Nicola Pitchford)
P (x, y) is a predicate with domain A implies {(a, b) | P (a, b) is true} is a relation on A. R1 and R2 are relations from A to B implies R1 ∪ R2 , R1 ∩ R2 , R1 − R2, and R1 are also. f is a function from A to B implies {(a, b) | a ∈ A ∧ f (a) = b} is a relation from A to B.
Politics have no relation to morals. (Niccolo Machiavelli)
A has n elements implies a relation on A has between 0 and n2 elements, 2 and implies 2n possible relations on A. CS 2233 Discrete Mathematical Structures
Relations Relations. . . . . . . . . . . . . . . . . . . . Examples of Relations . . . . . . . . . . . Types of Relations on A . . . . . . . . . Composite of Two Relations . . . . . . Transitivity and Composite Relations . Closures of Relations. . . . . . . . . . . . Transitive Closures and Paths. . . . . . A Transitive Closure Algorithm . . . . . Warshall’s Algorithm. . . . . . . . . . . . Other Types of Relations . . . . . . . . .
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Relations – 2
Examples of Relations Numeric comparisons: =, , 6= {(0, 0), (1, 1), (−1, −1), (2, 2), (−2, −2), . . .} √ Functions: log2 n, n, n2, 2n {(0, 0), (1, 1), (4, 2), (9, 3), (16, 4), . . .} Sequences: {(0, a0), (1, a1), . . .} {(0, 0), (1, 1), (2, 1), (3, 2), (4, 3), . . .} From From From From
names to ID numbers. zip codes to cities. countries to leaders. years to events.
CS 2233 Discrete Mathematical Structures
1
Relations – 3
2
Types of Relations on A
Transitivity and Composite Relations
R is reflexive iff (x ∈ A implies (x, x) ∈ R) Examples: =, ≤
R is transitive implies Rn ⊆ R for all n ≥ 1. Basis: R1 = R ⊆ R.
R is symmetric iff ((x, y) ∈ R implies (y, x) ∈ R) Examples: =, 6=
Induction: Assume k ≥ 1 and Rk ⊆ R. Show Rk+1 ⊆ R.
R is antisymmetric iff ((x, y) ∈ R implies x = y ∨ (y, x) 6∈ R) Examples: =, < R is transitive iff ((x, y) ∈ R ∧ (y, z) ∈ R implies (x, z) ∈ R) Examples: =,
