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Unformatted text preview: Relations with Applications to Cryptography Mariusz Bajger COMP2781/8781 School of Computer Science, Engineering and Mathematics May 24, 2011 1/27 Reading and Exercises Reading Epp, Chapter 8 Exercises Sec. 8.1 (skip inverse relation examples), Sec.8.2 (all except 3436, 4350), Sec. 8.3 (all), Sec. 8.4 (all) Good learning strategy: BE ACTIVE! I Regularly revise lectures I Solve the suggested exercises I Be critical when reading textbook/lectures I Ask your colleagues, ask the lecturer, don’t be shy! I It is OK to ask for help any time 2/27 Nary relations A binary relation R from a set X to a set Y is a subset of the Cartesian product X × Y . If X = Y we have a binary relation on X . Notation: x R Y or ( x , y ) ∈ R means that x is related to y . Given sets A 1 , A 2 ,..., A n , one can similarly define nary relation R on A 1 × A 2 ×···× A n as a subset of A 1 × A 2 ×···× A n . EXAMPLE: relational databases Student database record may look like (quaternary relation): ( a 1 , a 2 , a 3 , a 4 ) ∈ R ←− a student with ID number a 1 , name a 2 , enrolled on (date) a 3 , date of birth a 4 . 3/27 Relations may not be functions R = { ( x , y ) ∈ R × R  x 2 + y 2 = 1 } S = { ( x , y , z ) ∈ R × R × R  x 2 + y 2 + z 2 = 1 } Why not functions? Try x 2 + y 2 1 = 1 and x 2 + y 2 2 = 1. Does this imply y 1 = y 2 ? 4/27 Reflexivity, symmetry and transitivity ∀ x ∈ A , x R x For example: =, ≤ , ⊆ but not < , ⊂ , ̸ = ∀ x , y ∈ A , if x R y then y R x For example: =, ̸ = but not ⊂ or < ∀ x , y , z ∈ A , if x R y and y R z then x R z For example: < , =, ⊂ but not ̸ = 5/27 Directed graph of a relation Let A = { 2 , 3 , 6 , 7 , 8 , 9 } and define a relation R on A by x R y ⇐⇒ 3  ( x − y ) . Note how the symmetry, reflexivity and transitivity properties are visualized on the graph. 6/27 Transitive closure The transitive closure of a relation R is the relation R on A such that (1) R is transitive (2) R ⊆ R (3) R is the smallest relation satisfying (1) and (2), that is, if S any other relation satisfying (1) and (2) then R = S . EXAMPLE Find the transitive closure for relations R and P defined on the set A = { 2 , 3 , 6 , 7 , 8 , 9 } . x R y ⇐⇒ 3  ( x − y ) . and P is defined by P = { (2 , 2) , (6 , 7) , (8 , 8) , (7 , 8) } . 7/27 Equivalence relations Definition A relation R on a set A is called an equivalence relation if, and only if, R is reflexive, symmetric, and transitive. For each a ∈ A we define [ a ], the equivalence class of a by [ a ] := { x ∈ A  x R a } ....
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 Spring '08
 WOUTERS
 Algebra, Cryptography

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