Prove - digraph of a partial order has no cycle more than 1, Mathematics

Assignment Help:

Prove that the Digraph of a partial order has no cycle of length greater than 1.

Assume that there exists a cycle of length n ≥ 2 in the digraph of a partial order ≤ on a set A. This entails that there are n distinct elements a1 , a2 , a3 , ..., an like that a1 ≤ a2 , a2 ≤ a3 , ..., an-1 ≤ an and an ≤ a1 . Applying the transitivity n-1 times on a1 ≤ a2 , a2 ≤ a3 , ..., an-1 ≤ an , we get a1 ≤ an .As relation ≤ is anti-symmetric a1 ≤ an and an ≤ a1 together entails that a1 = an . This is contrary to the fact that all a1, a2, a3... an are distinct. So, our assumption that there is a cycle of length n ≥ 2 in the digraph of a partial order relation is wrong.

 


Related Discussions:- Prove - digraph of a partial order has no cycle more than 1

Size of the penumbra, With reference to Fig. 1(a) show that the magnificati...

With reference to Fig. 1(a) show that the magnification of an object is given by M=SID/SOD. With reference to Fig. 1(b) show that the size of the penumbra (blur) f is given by f

Solid mensuration, The two sides of a triangle are 17 cm and 28 cm long, an...

The two sides of a triangle are 17 cm and 28 cm long, and the length of the median drawn to the third side is equal to 19.5 cm. Find the distance from an endpoint of this median to

What is the greatest value of the number, Five more than the quotient of a ...

Five more than the quotient of a number and 2 is at least that number. What is the greatest value of the number? Let x = the number. Notice that quotient is a key word for div

Engg maths, How to get assignment to solve and earn money

How to get assignment to solve and earn money

Service marketing, assignment of marketing mix on healthservices

assignment of marketing mix on healthservices

1trig, how do you find the tan, sin, and cos.

how do you find the tan, sin, and cos.

Draw the digraph for the partial order, 1. Consider the relation on A = {1,...

1. Consider the relation on A = {1, 2, 3, 4} with relation matrix: Assume that the rows and columns of the matrix refer to the elements of A in the order 1, 2, 3, 4. (a)

Math, what is quantity ?

what is quantity ?

My daugther needs help, my daughter is having trouble with math she cant un...

my daughter is having trouble with math she cant understand why please help us

Intrgers, how to evaluate the sums

how to evaluate the sums

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd