DISCRETE STRUCTURES For each of the following relations defi

DISCRETE STRUCTURES:

For each of the following relations defined on the set of all students, determine whether it is reflexive, symmetric, and transitive.
(a) aRb if a is strictly older than b.
(b) aRb if a and b have the same first name.
(c) aRb if a and b have played the same sport at the high school or collegiate level.
Which of the above relations is an equivalence relation? Describe the equivalence class that contains YOU.

Solution

(a)

a is NOT strictly older than a => a is NOT related to a => R is NOT reflexive

if aRb,then a is strictly older than b =>b is NOT strictly older than a =>b is NOT related to a

=> R is NOT symmetric

if aRb, bRc , then a is strictly older than b, b is strictly older than c => a is strictly older than c =>a Rc

=>

R is transitive

R is NOT equivalence relation

(b)

a and a have the same first name => aRa => R is reflexive

if aRb, then a and b have same first name => b and a have same first name => bRa => R is symmetric

if aRb, bRc, then a and b have same first name, b and c have same first name => a and c have same first name

=> aRc => R is transitive

=> R is Equivalence relation

(c)

a and a have played the same sport at high school or collegiate level => aRa => R is reflexive

if aRb, then a,b played the same sport at high school or collegiate level

=>

b,a played the same sport at high school or collegiate level => bRa => R is symmetric

if aRb, bRc, a,b then played the same sport at high school or collegiate level, b,c played the same sport at high school or collegiate level

=>

a,c played the same sport at high school or collegiate level => aRc => R is transitive

=>

R is equivalence relation