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

a) R is reflexive, R is not symmetric, R is not anti-symmetric, R is transitive.

b) R is reflexive, R is symmetric, R is not anti-symmetric, R is not transitive.

c) R is not reflexive, R is symmetric, R is not anti-symmetric, R is not transitive.

Step-by-step explanation:

a)

(a, b) ∈ R if and only if everyone who has visited Web page a has also visited Web page b.

Obviously R is reflexive (aRa)

Everyone who has visited Web page a has also visited Web page a

R is not symmetric (aRb does not imply bRa)

If everyone who has visited Web page a has also visited Web page b does not mean that everyone who has visited Web page b has also visited Web page a

R is not anti-symmetric (aRb and bRa does not imply a=b)

If everyone who has visited Web page a has also visited Web page b and everyone who has visited Web page b has also visited Web page a does not mean the web pages are the same.

R is transitive (aRb and bRc implies aRc)

If everyone who has visited Web page a has also visited Web page b and everyone who has visited Web page b has also visited Web page c implies that everyone who has visited Web page a has also visited Web page c.

b)

(a, b) ∈ R if and only if there are no common links found on both Web page a and Web page b.

R is obviously reflexive (aRa)

R is symmetric (aRb implies bRa)

if there are no common links found on both Web page a and Web page b, then there are no common links found on both Web page b and Web page a.

R is not anti-symmetric (aRb and bRa does not imply a=b)

if there are no common links found on both Web page a and Web page b and there are no common links found on both Web page b and Web page a does not mean a and b are the same web page.

R is not transitive (aRb and bRc does not imply aRc)

Consider for example three web pages a, b and c such that a and c have a common link and b has no external links at all.

Then obviously (a, b) ∈R and (b, c) ∈R since b has no links, but (a, c) ∉R because they have a common link.

c)

(a, b) ∈ R if and only if there is at least one common link on Web page a and Web page b

R is not reflexive

If the web page a does not have any link at all, then a is not related to a.

R is symmetric (aRb implies bRa)

if there is at least one common link found on Web page a and Web page b, then there is at least one common link found on Web page b and Web page a.

R is not anti-symmetric (aRb and bRa does not imply a=b)

if there is at least one common link found on Web page a and Web page b and there is at least one common link found on Web page b and Web page a does not mean the web pages are the same

R is not transitive (aRb and bRc does not imply aRc)

Consider for example three web pages a, b and c such that a has only two links L1 and L2, b has only two links L2 and L3 c has only two links L3 and L4.

Then (a, b) ∈ R since a and b have the common link L2, (b, c) ∈ R for b and c have the common link L3, but a and c have no common links, therefore (a, c) ∉R