Discrete Structure

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# Discrete Structure - PowerPoint PPT Presentation

Discrete Structure. Li Tak Sing( 李德成 ). Chapter 4 Properties of Binary Relations. Three special properties For a binary relation R on a set A, we have the following definitions. R is reflexive if xRx for all x A. R is symmetric if xRy implies yRx for all x,y A

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### Discrete Structure

Li Tak Sing(李德成)

Chapter 4Properties of Binary Relations
• Three special properties
• For a binary relation R on a set A, we have the following definitions.
• R is reflexive if xRx for all xA.
• R is symmetric if xRy implies yRx for all x,y A
• R is transitive if xRy and yRz implies xRz for all x,y,z A
Two opposite properties
• For a binary relation R on a set A, we have the following definitions.
• R is irreflexive if (x,x)R for all xA.
• R is antisymmetric if xRy and yRx implies x=y for all x,yA.
Example
• R is a binary relation on N
• aRb if (a+b) mod 2 = 0
• R is reflexive because (a+a) mod 2 =0 for all a N
• R is symmetric because if aRb, then (a+b) mod 2 = 0, then (b+a) mod 2 =0, then bRa
• R is transitive, because if aRb and bRc, then (a+b) mod 2 =0 and (b+c) mod 2 =0, then (a+2b+c) mod 2 =0, then (a+c) mod 2 =0, then aRc
Example
• Give examples of binary relations over the set {a,b,c,d} with the stated properties:
• Reflexive and not symmetric and not transitive
• Symmetric and not reflexive and not transitive
• transitive and not reflexive and not symmetric
Composition of relations
• If R and S are binary relations, then the composition of R and S, which we denote by RS, is the following relation:RS={(a,c)|(a,b)R and (b,c) S for some element b}
More examples
• For each of the following binary relations state which of the three properties, reflexive, symmetric and transitive are satisfied.
• xRy iff |x-y| is odd, over the integers.
• xRy iff x is a parent of y, over the set of people.
Grandparents
• Given the isParentOf relation. So a isParentOf b represents the fact that a is the parent of b.
• isGrandparentOf can then be defined in terms of isParentOf.isGrandparentOf=isParentOfisParentOf
• So a isGrandparentOf b if there is c so that a isParentOf c and c isPrentOf b.
More examples
• Given the following binary relations over {a,b,c,d}.R={(a,a),(a,c),(b,a),(b,d),(c,b)}S={(a,b),(a,c),(c,b),(d,c)}
• Find RS
• Find SR
Representations
• If R is a binary relation on A, then we'll denote the composition of R with itself n times by writing Rn.
• For example,
• isGrandparentOf=isParentOf2
• isGreatGrandParentOf=isParentOf3
Inheritance properties
• If R is reflexive, then Rn is reflexive.
• If R is symmetric, then Rn is symmetric.
• If R is transitive, then Rn is transitive.
Example
• Let R={(x,y)ZZ|x+y is odd}. We want to find out R2 and R3.
Closures
• If R is a binary relation and p is some property, then the p closure of R is the smallest binary relation containing R that satisfies property p.
Reflexive closure
• A reflexive closure of R is the smallest reflexive relation that contains R.
• A reflexive closure of R is denoted as r(R)
• R is a relation over {a,b,c} and R={(a,b),(b,c)}Then, r(R)={(a,a),(b,b),(c,c),(a,b),(b,c)}
Symmetric closure
• A symmetric closure of R is the smallest symmetric relation that contains R.
• A symmetric closure of R is denoted as s(R)
• R={(a,b),(b,c)}, s(R)={(a,b),(b,a),(b,c),(c,b)}
Transitive closure
• A transitive closure of R is the smallest transitive relation that contains R. It is denoted as t(R).
• R= ={(a,b),(b,c)}, then t(R)= {(a,b),(b,c),(a,c)}
Constructing Closures
• If R is a binary relation over a set A, then:
• r(R)=RRo (Ro is the equality relation)
• s(R)=R Rc (Rc is the converse relation)
• t(R)=R R2 R3 R4....
• If A is finite with n elements, then t(R)= R R2 R3 R4.... Rn
Example
• Given the set A={a,b,c,d}. Draw a directed graph to represent the indicated closure for each of the following binary relations over A.
• r(R), where R={(a,d)}
• s(R) where R={(a,b), (c,d)}
• t(R) where R={(a,b),(d,a),(d,c),(c,b)}