# Discrete Mathematics Basics 1) Determine whether the relation R on the set of all Web

Discrete Mathematics Basics
1) Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where $\left(a,b\right)\in R$ if and only if
I) everyone who has visited Web page a has also visited Web page b.
II) there are no common links found on both Web page a and Web page b.
III) there is at least one common link on Web page a and Web page b.
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Step 1
We are entitled to solve only the first 3 subparts of the first question so, I am providing you the same. Kindly repost the question to get the desired help.
Step 2
The relation is reflexive if every element in A, (a,a) exists in the relation.
The relation is symmetric if $\left(a,b\right)\in R$ whenever $\left(b,a\right)\in R$.
The relation is antisymmetric if the existence of (a,b) and (b,a) indicates that $a=b$.
The relation is trnasitive if $\left(a,b\right)\in R$ and $\left(b,c\right)\in R$ then $\left(a,c\right)\in R$.
Step 3
Assume $A=$ set of all webpages
I) If a person has visited webpage A then he has visited webpage. So, $\left(a,a\right)\in R$ for every element in A. Thus, the relation is reflexive.
It is possible that some people has visited webpage B but not has visited webpage A. So, the relation is not symmetric.
If everyone who has visited webpage A has also visited webpage B and If everyone who has visited webpage B has also visited webpage A, then these webpages are equal. So, the relation is not antisymmetric.
If everyone who has visited webpage A has also visited webpage B and If everyone who has visited webpage B has also visited webpage C, then everyone who has visited webpage A has also visited webpage C. Thus, the relation is transitive.
Step 4
II) The relation will always have a common link to itself. So, the relation is not reflexive.
If webpage A and webpage B have no common link then, webpage B and webpage A will also have no common link. So, the relation is symmetric.
If webpage A and webpage B has no common link and webpage B and webpage A has no common link, then it is not necessary that these webpages are equal. So, the relation is not anitsymmetric.
If webpage A and webpage B has no common link and webpage B and webpage C has no common link, then it is possible that webpage A and webpage B has some common link. So, the relation is not transitive.
Step 5
III) All elements in A will have not common link with itself. So, the relation is not reflexive.
If webpage A and webpage B has one common link, then webpage A and webpage B will also has one common link. So, the relation is symmetric.
If webpage A and webpage B has one common link and webpage B and webpage A has one common link, then it is not necessary that these webpages are equal. So, the relation is not antisymmetric.
If webpage A and webpage B has one common link and webpage B and webpage C has one common link, then it is not necessary that webpage A and webpage C has one common link. So, the relation is not transitive.