Determine whether the relation is reflexive, symmetric, anti-symmetric, and/or transitive?

Question

Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if

a) everyone who has visited Web page a has also visited Web page b.

b) there are no common links found on both Web page a and Web page b.

c) there is at least one common link on Web page a and Web page b.

d) a is taller than b.

e) a and b were born on the same day.

f) a has the same first name as b.

I have the knowledge about the reflexive, symmetric, anti-symmetric, and transitive also I can able to solve the problems when the  relations are defined in ordered pairs. But can't able to visualize this kind of relations.

Comments

you can read such kind of problems.Hope it will help you.

https://cseweb.ucsd.edu/classes/wi08/cse20/week8prob1_2.pdf

https://drive.google.com/file/d/1Pd6-t9K7qjFOK9mvHX2AE0inkK7Z-uuh/view?usp=sharing 

Answer 1

a) is Reflexive( because everyone who has visited web page a has also visited web page a), Not Symmetric( because if everyone who has visited web page a has visited web page b, this does not mean that everyone who has visited web page b has also visited web page a),  Not antisymmetric(because if everyone who has visited web page a has visited web page b and everyone has visited web page b has visited web page a, it does not necessarily mean that a and b are the same web page), Transitive( because if everyone who has visited web page a also has visited web page b and everyone who has visited web page b also has visited web page c, then it follows that everyone who has visited web page aa has also visited web page c)

b) Relation R would be something like R(a,b) = Linkoutof(a) intersection Linkoutof(b) = NULL 

 Not Reflexive, Symmetric, Not AntiSymmetric: (two distinct pages may have no common links), NOT Transitive.

c) Relation R would be something like R(a,b) = Linkoutof(a) intersection Linkoutof(b) != NULL 

 Not Reflexive: (some pages have no outgoing links), Symmetric, Not AntiSymmetric: (two distinct pages  may be have a  common links), Transitive.

d) is Not reflexive(because a cannot be taller than itself), Not symmetric(X is taller than me. I am not taller than X), Transitive, AntiSymmetric(clearly)

e) is reflexive, symmetric, transitive, not antisymmetric (Two distinct people may be born on the same day)

f) is  reflexive, symmetric, transitive, not antisymmetric (Two distinct people may have same first name)

Answer 2

1.  a is taller than b. Looks like  A>=B now it is reflexive , antisymmetric ,and transitive.

example : a= 10;  b= 3 or 10 then 10>=3 true so antisymmetric 

                  10=10 so reflexive.

2. a and b were born on the same day. Looks like A=B now it is reflexive, symmetric, transitive , anitisymmetric.