Determine whether the relation R on the set of all Web pages is reflexive, symme

Question

Determine whether the relation R on the set of all Web pages is reflexive, symmetric, anti-symmetric, 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.

d) There is a Web page that includes links to both Web page a and Web page b.

Determine whether the relation R on the set of all Web pages is reflexive, symmetric, anti-symmetric, and/or transitive, where (a, b) R if and only if: Everyone who has visited Web page a has also visited Web page b. There are no common links found on both Web page a and Web page b. There is at least one common link on Web page a and Web page b. d) There is a Web page that includes links to both Web page a and Web page b.

Explanation / Answer

(a)

Reflexive (since if someone visit page a, then he also visit page a :-))

Not symmetric (If someone visit a then b, it doesn't mean that if visit b he'll visit a)

Not antisymmetric (If someone visit a, then b and b then a, it doesn't mean that a=b)

Transitive , since if someone visit a, then b and b then c, then if he visit a he'll visit c

(b)

Not reflexive (Page a and Page doesn't have empty intersection of links (unless they have no link)

Symmetric ( links(a) n links(b) = links(b) n links(a) = empty)

Not antisymmetric (Two distinct pages can have no commun link)

Not transitive ( Think with set A={1},B={2},C={1,3} , then AnB=empty,BnC=empty but AnC={1})

(c)

Not reflexive (Think of pages without outgoing links)

Symmetric ( Links(a) n Links(b) = Links(b) n Links(a) != empty )

Not antisymmetric (Two discinct page can have a commun link)

Not transitive ( Think with set A={1},B={1,2},C={2,3} , then AnB={1} (not empty),BnC={2} (not empty) but AnC=empty)

(d)

Not reflexive (if we assume there are pages with no link pointing to them, otherwise reflexive)

Symmetric (If a webpage includes a link to a and b, then he includes a link to b and a)

Not antisymmetric (If a webpage includes a link to a and b (and b and a), then it doesn't mean that a=b)

Not transitive (A webpage can includes a link to a and b and another to b and c, the page might be different so it doesn't garantee a link to a and c in the same page somewhere)

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