July 2020
Intermediate to advanced
384 pages
11h 25m
English
Content preview from A Modern Introduction to Fuzzy Mathematics
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.
5Possibility Theory
The English economist George Lennox Sharman Shackle [256] proposed that uncertainty is related to surprise. For example, when Jane says that an event can happen, then she will not be surprised if this event will actually happen. In general, when we say that an event can “happen”, we actually mean that it is a “possible” event. Since there are degrees of surprise, it is absolutely reasonable to talk about degrees of possibility. Thus, in a way, Shackle laid the theoretical basis for a theory of possibility. Indeed, Zadeh [314] used fuzzy sets to define a mathematical theory of possibility. Later on, others tried to describe possibilities using different tools that culminated to what is now known as possibility theory.
5.1 Fuzzy Restrictions and Possibility Theory
Although Zadeh was quite aware of the measure‐theoretic axiomatization of probability1 (see Ref. [257] for a discussion of Kolmogorov's work), still he opted to define the notion of possibility in terms of fuzzy restrictions [309]. In Zadeh's own words, a fuzzy restriction is “a fuzzy relation which acts as an elastic constraint on the values that may be assigned to a variable.” Here the term “elastic” does not have any special meaning, and it just means that the constraint is not rigid. Let us first give an informal description of fuzzy restrictions.
Consider the fuzzy proposition Emma is young and let denote the value of variable Emma. The value is a whole number that belongs to the interval ...
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Read now
Unlock full access
More than 5,000 organizations count on O’Reilly
O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.Julian F.
I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.Addison B.
I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.Amir M.
I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.