Given two binary fuzzy relations R and M$,$ R : T $$ D and M : T $$ T describe a relevance relation and a fuzzy thesaurus, respectively. Rij describes a degree of relevance of given index terms ti $$T to a given document dj $$ D$,$ while Mkl describes the degree of similarity in the meaning of the index terms tk and tl$.$

Consider the set D $=\{$d$1,$ d$2,...,$ d$5\}$

of only $10$ documents, and the set

T $=\{$t$1,$ t$2,...,$ t$5\}$

of the following index terms:

t$1$: fuzzy number

t$2$: fuzzy arithmetic

t$3$: fuzzy relation

t$4$: fuzzy set

t$5$: fuzzy proposition

Assume that the relevance of these index terms to the individual documents in set D has been determined and is expressed by the matrix R below:

R d$1$ d$2$ d$3$ d$4$ d$5$

t$10\mathrm{.}200\mathrm{.}701$

t$200\mathrm{.}300\mathrm{.}91$

t$300\mathrm{.}6010\mathrm{.}5$

t$400\mathrm{.}3000$

t$50\mathrm{.}410\mathrm{.}80\mathrm{.}51$

Also assume that the similarity in meanings of the index terms has been expressed by the matrix M below:

M t$1$ t$2$ t$3$ t$4$ t$5$

t$110\mathrm{.}510\mathrm{.}30\mathrm{.}8$

t$20\mathrm{.}510\mathrm{.}70\mathrm{.}10$

t$310\mathrm{.}7100\mathrm{.}2$

t$40\mathrm{.}30\mathrm{.}1010\mathrm{.}9$

t$50\mathrm{.}800\mathrm{.}20\mathrm{.}91$

In fuzzy information retrieval, any user$$s inquiry may be expressed by a fuzzy set Ion T $,$ in which the membership grade is a degree of the possible user$$s inquiry.

I $=\{0\mathrm{.}3/$t$1,0\mathrm{.}7/$t$2,1/$t$3,0\mathrm{.}8/$t$4,0\mathrm{.}9/$t$5\}$

$1)[10]$ An augmented inquiry, A$,$ may be extended from the original inquiry by associated index terms:

A $=$ I $$ M$.$

Compute A with an augmented inquiry of a user I from the given relations matrices above.

$2)[10]$ The retrieved documents are characterized by a fuzzy set Q on D$,$ by composing the augmented inquiry A with the relevance relation A$,$ Q $=$ A $$ R$.$ Compute Q for a user I.

$3)[5]$ The set Q characterizes the degree to which each document matches with the user$$s interest as expressed by her$/$his inquiry. What document is I$$s the most interest.