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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 xA.
- 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 xA.
- R is antisymmetric if xRy and yRx implies x=y for all x,yA.

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 RS, is the following relation:RS={(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=isParentOfisParentOf
- 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 RS
- Find SR

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)ZZ|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)=RRo (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)}

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