sanuluy

2021-08-18

Discrete Mathematics Basics

1) Find out if the relation R is transitive, symmetric, antisymmetric, or reflexive on the set of all web pages.where $\left(a,b\right)\in R$ if and only if
I)Web page a has been accessed by everyone who has also accessed Web page b.
II) Both Web page a and Web page b lack any shared links.
III) Web pages a and b both have at least one shared link.

l1koV

Step 1
Only the first three subparts of the first question may be solved, so I am giving you those. Please re-post your query if you still need assistance.
Step 2
If each element of A, (a,a) exists in the relation, the relation is reflexive.
The relation is symmetric if $\left(a,b\right)\in R$ whenever $\left(b,a\right)\in R$
If the existence of (a,b) and (b,a) shows that $a=b$, then the connection is antisymmetric. .
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) An individual has viewed webpage B if they have visited webpage A.. So, $\left(a,a\right)\in R$ for every element in A. Thus, the relation is reflexive.
It's feasible that some individuals have visited website B but not website A. The relationship is hence asymmetric.
If everyone who has visited webpage A has also visited webpage B, and vice versa, then these two webpages are equal. The relationship is therefore not asymmetric.
If every person who visited webpage A also visited webpage B, and every person who visited webpage B also visited webpage C, then every person who visited webpage A also visited webpage C. The relationship is hence transferable.
Step 4
II) The relation will always be connected to itself in some way. The relation is therefore not reflexive. If webpage A and website B don't share a link, then neither will webpage B and webpage A. The relation is thus symmetrical.
It is not necessary for these webpages to be equal if webpage A and webpage B share no links and webpage B and webpage A share no links. The relationship is therefore not asymmetric.
If webpages A and B do not share a link and webpages B and C do not share a link, then it is feasible that webpages A and B do share a link. The relationship is therefore not transferable.
Step 5
III) Each element in A will be distinct from the others. The relationship is therefore not reflexive.
If webpages A and B share a link, then those two webpages will also share that connection. The relationship is thus symmetrical.
It is not essential for these webpages to be equal if webpage A and webpage B share a link and webpage B and webpage A share a link. The relationship is therefore not asymmetric.
It is not necessary for webpage A and webpage C to share a link if webpage B and webpage C share a link and webpage A and webpage B share a link. The relationship is therefore not transferable.

RizerMix

- R is transitive.
- R is symmetric.
- R is not antisymmetric.
- R is not reflexive.
Explanation:
I) To check transitivity, we need to verify if whenever (a,b) and (b,c) are in R, then (a,c) must also be in R.
For this condition, if Web page a has been accessed by everyone who has also accessed Web page b, and Web page b has been accessed by everyone who has also accessed Web page c, then it implies that Web page a has been accessed by everyone who has also accessed Web page c. Thus, R is transitive.
II) To check symmetry, we need to determine if whenever (a,b) is in R, then (b,a) must also be in R.
For this condition, if both Web page a and Web page b lack any shared links, it implies that Web page b also lacks any shared links with Web page a. Thus, R is symmetric.
III) To check antisymmetry, we need to verify if whenever (a,b) and (b,a) are in R, then a must be equal to b.
For this condition, if Web pages a and b both have at least one shared link, it does not guarantee that they are the same web page. Therefore, R is not antisymmetric.
IV) To check reflexivity, we need to determine if for every web page a, (a,a) is in R.
Since none of the conditions explicitly state that every web page must be accessed by itself or lack shared links with itself, we cannot conclude that R is reflexive.

Vasquez

To analyze the properties of the relation R on the set of all web pages, let's consider each condition individually.
I) For the relation (a, b) ∈ R if and only if ''Web page a has been accessed by everyone who has also accessed Web page b.''
To determine if R is reflexive, we need to check if every web page has been accessed by everyone who has accessed itself. In other words, for all web pages 'a,' we need to verify if (a, a) ∈ R holds true. However, since it is not specified in the condition that every web page is accessed by itself, we cannot conclude that R is reflexive.
To determine if R is symmetric, we need to check if whenever (a, b) ∈ R, then (b, a) ∈ R also holds true. According to the given condition, if web page 'a' has been accessed by everyone who has also accessed web page 'b,' it does not necessarily imply that web page 'b' has been accessed by everyone who has accessed web page 'a.' Therefore, R is not symmetric.
To determine if R is antisymmetric, we need to check if whenever (a, b) ∈ R and (b, a) ∈ R, then a = b. From the given condition, we can see that if web page 'a' has been accessed by everyone who has also accessed web page 'b' and vice versa, it implies that both web pages have the same set of users who accessed them. Therefore, R is antisymmetric.
To determine if R is transitive, we need to check if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R. In this case, if web page 'a' has been accessed by everyone who has also accessed web page 'b,' and web page 'b' has been accessed by everyone who has also accessed web page 'c,' it implies that web page 'a' has been accessed by everyone who has accessed web page 'c.' Therefore, R is transitive.
II) For the relation (a, b) ∈ R if and only if ''Both Web page a and Web page b lack any shared links.''
Since the condition states that both web pages 'a' and 'b' lack any shared links, we can immediately conclude that no web pages share links. Therefore, R is reflexive, symmetric, antisymmetric, and transitive.
III) For the relation (a, b) ∈ R if and only if ''Web pages 'a' and 'b' both have at least one shared link.''
To determine if R is reflexive, we need to check if every web page has at least one shared link with itself. However, since it is not specified in the condition that every web page has at least one shared link with itself, we cannot conclude that R is reflexive.
To determine if R is symmetric, we need to check if whenever (a, b) ∈ R, then (b, a) ∈ R also holds true. From the given condition, if web pages 'a' and 'b' both have at least one shared link, it implies that web pages 'b' and 'a' also have at least one shared link. Therefore, R is symmetric.
To determine if R is antisymmetric, we need to check if whenever (a, b) ∈ R and (b, a) ∈ R, then a = b. According to the given condition, if web pages 'a' and 'b' both have at least one shared link, it does not imply that they are the same web page. Therefore, R is not antisymmetric.
To determine if R is transitive, we need
to check if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R. In this case, if web pages 'a' and 'b' both have at least one shared link, and web pages 'b' and 'c' also have at least one shared link, it implies that web pages 'a' and 'c' have at least one shared link. Therefore, R is transitive.
In summary:
I) R is antisymmetric and transitive, but not reflexive or symmetric.
II) R is reflexive, symmetric, antisymmetric, and transitive.
III) R is symmetric and transitive, but not reflexive or antisymmetric.

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