I get easily confused when it comes to the symbol based terminology in Set Theory. Could someone please elaborate on what the following expressions mean? It would really help me out.
- $S_1$ = knowledge of a subject matter
- $S_2$ = problem solving related to this subject matter
$S_3$ = ability to adapt properly the already existing knowledge for use in analogous similar cases
set $U = \{a, b, c, d, e\}$.
- set MAi = subset of U
Denote by $a, b, c, d$, and $e$ the linguistic labels (fuzzy expressions) of very low, low, intermediate, high and very high success respectively of a student in each of the $S_i$s and set $U = \{a, b, c, d, e\}$.
Question 1 What exactly is $S_i$s? I realize the $i$ is a subscript and the $S$ represents the 'array' of characteristics. But what's up with the trailing s?
There is another statement that says:
We define the membership function $m_{A_i}$ for each $x$ in $U$ as ...
What exactly is $x$? Since the elements are not integers, would $x$ be the index number and therefore represent $a$ as $0$, $b$ as $1$ and so on?
I am terribly sorry about asking these really basic questions. I have gaps in the fundamentals of my knowledge which are hampering me. I would be very grateful if someone could help me bridge them.
Thanks.
‘$S_i$s’ is just the plural of $S_i$: the $S_i$s are the characteristics $S_1,S_2$, and $S_3$.
In the second question, $x$ ranges over the set $U$: its possible values are $a,b,c,d$, and $e$. The objects $A_1,A_2$, and $A_3$ aren’t ordinary sets; they’re so-called fuzzy sets. Instead of objects being definitely in or definitely not in them, objects have some degree of membership between $0$ and $1$.
The idea is that for $i=1,2,3$ the fuzzy subset $A_i$ of $U$ represents the distribution of students who scored very low, low, intermediate, high, and very high on characteristic $S_i$. If over $80\%$ of the group scored at level $x$ (where $x$ is one of $a,b,c,d,e$), then $m_{A_i}(x)=1$; if more then $60\%$ but at most $80\%$ scored at level $x$, then $m_{A_i}(x)=0.75$, and so on, according to the definition of the function on the third page of the PDF. For example, if $20\%$ of the students scored very low on $S_1$, $30\%$ scored intermediate, and the remaining $50\%$ very high, then the membership function for the fuzzy set $A_1$ would be
$$f(x)=\begin{cases} 0,&\text{if }x=a\\ 0,&\text{if }x=b\\ 0.25&\text{if }x=c\\ 0,&\text{if }x=d\\ 0.5,&\text{if }x=e\;, \end{cases}$$
since $20\%$ is no more than $\frac15$, $30\%$ is between $\frac15$ and $\frac25$, and $50\%$ is between $\frac25$ and $\frac35$.