Page 30 - Discrete Structure II

b) R = {(1,1), (1,2), (2,1), (2,2), (3,3), (4,4) }

Reflexive yes
Symmetric yes
Antisymmetric no (1,2) , (2, 1)  R and 1  2
Transitive yes

c) R = { (2,4), (4,2)}
Reflexive no (1,1)  R
Symmetric yes
Antisymmetric No (2,4) , (4, 2)  R and 2  4
Transitive No (2,4) , (4, 2)  R and (2,2)  R

d) R = {(1,2), (2,3), (3,4) }
Reflexive No (1,1)  R
Symmetric No (1,2)  R and (2,1)  R
Antisymmetric yes
Transitive No (1,2) , (2, 3)  R and (1,3)  R

e) R = {(1,1), (2,2), (3,3), (4,4) }
Reflexive yes
Symmetric yes
Antisymmetric yes
Transitive yes

f) R = {(1,3), (1,4), (2,3), (2,4), (3,1), (3,4) }
Reflexive No (1,1)  R
Symmetric No (1,4)  R and (4,1)  R
Antisymmetric No (1,3) , (3, 1)  R and 1  3
Transitive No (1,3) , (3, 1)  R and (1,1)  R

n-ary Relations with their Applications

Introduction

Recall a relation binary from a set A to a set B is a subset of the Cartesian product A x B.

R is a binary relation from A to B if R = {(a, b) such that a is an element of A and b is an element
of B}

Example:

A ={a, b} B = { a, c, d}

A x B = { (a, a), (a, c), (a, d), (b, a), (b, c), (b, d)} six elements