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
