I am reading the book "Fuzzy Logic with Engineering" written by Timothy J. Ross and there is a phrase in the book at page 90, he mentions about the "Features of the Membership Function". He says:

The support of a membership function for some fuzzy set A under ~ is defined as that region of the universe that is characterized by nonzero membership in the set A under ~. That is, the support comprises those elements x of the universe such that μ A under ~ (x)>0.

He denotes it as a graphical representation:

enter image description here

what is μ A under ~ (x) in this context? Thanks in advance.

Edit Post The μ A under ~ (x) in this context is membership function. https://www.tutorialspoint.com/fuzzy_logic/fuzzy_logic_membership_function.htm

  • $\begingroup$ The picture is pretty clear: the support of $A$ is the set of $x$ such that $\mu(x) > 0$. The core is the set of $x$ such that $\mu(x) = 1$, and the boundary is the set of $x$ such that $0 < \mu(x) < 1$. $\endgroup$ – Yuval Filmus Apr 5 '20 at 13:59
  • $\begingroup$ What is the definition of μ? $\endgroup$ – tahasozgen Apr 5 '20 at 14:16
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    $\begingroup$ I have no idea, since I don't have access to the book. However, I am certain that $\mu$ is defined in the book, moreover in the first 90 pages. $\endgroup$ – Yuval Filmus Apr 5 '20 at 14:17

Here μ A˜(∙) μA~(∙) membership function of A˜ this assumes values in the range from 0 to 1, i.e., μ A ˜ (∙)∈[0,1] μA~(∙)∈[0,1] . The membership function μ A ˜ (∙) μA~(∙) maps U to the membership space M . The dot (∙) in the membership function described above, represents the element in a fuzzy set; whether it is discrete or continuous.

In classical set theory, it is either 1 or 0.




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