**Things you need to know**

Refer to short notes : http://www.techtud.com/short-notes/basics-relation

Now,

**a)** __everyone who has visited Web page a has also visited Web page b.__

(I) __Reflexive__: (a, a) ∈ R

because, everyone who has visited web-page **a** has by default visited **a **

** ** (II) __Transitive__: refer to the def. of Transitive and example on http://www.techtud.com/short-notes/basics-relation

(III)__ NOT Symmetric__: Because, A/Q (a,b) ∈ R, but (b,a) ∉ R,

(IV)** **__NOT Antisymmetic__ : Because, A/Q (a,b) ∈ R, but (b, a) can belong to R for some of webpages..

What this means ?

Consider WebPage = {1,2,3}

and R = { (1,1), (2,2), (3,3), (1, 2 ) , (2,1), (1,3) , (3,2)}

This relation is antisymetric despite the fact that (1,2) and (1,2) both ∈ R

**(b) **__ ____there are no common links found on both Web page a and Web page b __

(I) __NOT Reflexive__ : As note the relation says that "there are no common links found on both Web page a and Web page b " , i.e if (a, b) ∈ R,

a, b do not share common links ∴ (a,a) ∉ R, hence NOT Reflexive.

(II) __Symmentric__ : Symmetric says that if (a,b) ∈ R, then (b,a ) ∈, this is True for the given relation.

(III)__NOT Transitive:__ it says that (a, b) ∧ (b, c) ⇒ (a , c) , but if this holds then web-page **a** and web-page **b **will have a common link to web- page **c**

** **(IV) __NOT Asymmetric:__ as it is symmetric

**(c) **__there is at least one common link on Web page a and Web page b.__

(I) __NOT Reflexive__ : (a,b) ∈ R, if a and b share at-least one common link, but consiter a web page with no links (e) then (e,e) ∉ R. Therefore not reflexive

(II) __Symmetric__ : if (a, b) ∈ R, then a and b share atleast one common link, therefore (b,a) ∈ R. Hence , Symmetric

(III)__ NOT Transitive:__ if (a, b) ∈ and (b, c) ∈ R, then it is not necessary that (a,c) ∈ R. Because, if web-page a and web-page b share

at- least one common link and web-page b and web-page c share at-least one common link then, it is not necessary that web-page a and web-page will also share a common link.

(IV) __NOT Antisymmetric: __ as it is symmetric

**(d) **__there is a Web page that includes links to both Web page a and Web page b.__

(I) __NOT Reflexive:__ (a,b) ∈ R, if there exist a web-page (say web-page m) such that m has links for both web-page a and web-page b . Now,

(a, a) need not belong to R, as the could be no web-page pointing to web-page a.

(II) __Symmetric:__ if (a, b) ∈ R, i.e there is a web-page linking to web-page a and web-page b, then (b,a) ∈ R. Hence, Symmetric.

(III)__ NOT Transitive:__ If ( a, b) ∈ R, and (b, c) ∈ then it is not necessary that (a,c)∈ R. If web-page e points to web-page a and web-page b, web-page f points to web-page b and web-page c then it is not necessary that there exit a web-page which'll point to both web-page a and

web-page c.

(IV) __NOT Antisymmetric:__ As it is symmetric.