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

sanuluy 2021-08-18 Answered
Discrete Mathematics Basics
1) 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
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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Expert Answer

l1koV
Answered 2021-08-19 Author has 100 answers
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 (a,b)R whenever (b,a)R.
The relation is antisymmetric if the existence of (a,b) and (b,a) indicates that a=b.
The relation is trnasitive if (a,b)R and (b,c)R then (a,c)R.
Step 3
Assume A= set of all webpages
I) If a person has visited webpage A then he has visited webpage. So, (a,a)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.
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