Set Theory Symbols

Question

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.

It would help if you say where these (rather odd) bits of symbolism come from ... — Peter Smith, Nov 09 '13 at 12:06
@Brian: Spot on :) I am amazed that you located it so fast. Have you been through this article before? By the way, the questions pertain to page 3. Thanks! — SPI, Nov 09 '13 at 12:12
@DannyDan: Wouldn't that imply that there are multiple instances of the set denoted by the subscript? — SPI, Nov 09 '13 at 12:21
@DannyDan: Never mind, I see what you mean. I am not a math wiz but isn't the use of an s to denote a plural a little misleading? Especially since it isn't preceded/followed by an apostrophe? — SPI, Nov 09 '13 at 12:24

score 0 · Accepted Answer · answered Nov 09 '13 at 12:28

‘$S_i$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 $\frac35$.

Thank you for shedding light on this! I really appreciate it :) — SPI, Nov 09 '13 at 12:39

Set Theory Symbols

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