A bell shaped membership function is specified by three parameters (π,π,π)β¦
2015
A bell shaped membership function is specified by three parametersΒ (π,π,π)Β as follows:
- A.
\(\dfrac{1}{1+\bigg(\dfrac{x-c}{a} \bigg)^b} \\\) - B.
\(\dfrac{1}{1+\bigg(\dfrac{x-c}{a} \bigg)^{2b}} \\\) - C.
\(1+\bigg(\dfrac{x-c}{a}\bigg)^b \\\) - D.
\(1+\bigg(\dfrac{x-c}{a} \bigg)^{2b}\)
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Correct answer: B
Answer: ΞΌ(x) = 1 / (1 + ((x - c)/a)^(2b))
Key properties of this form:
Peak at x = c: when x = c the powered term is 0, so ΞΌ(c) = 1/(1+0) = 1.
Tails go to zero: as |x - c| β β the powered term β β, so ΞΌ(x) β 0, producing bell-shaped tails.
Even exponent ensures symmetry and nonnegativity: using (2b) makes ((x - c)/a)^(2b) β₯ 0 for all x, so the curve is symmetric about c and well defined for real-valued b.
Parameter interpretations: c is the center (location of peak), a scales the width (larger a β wider bell), and b controls steepness (larger b β steeper sides).
Why the other presented expressions are not appropriate:
Expression with exponent b but without squaring (1/(1 + ((x - c)/a)^b)): may not be symmetric or well defined for non-integer b because the base ((x - c)/a) can be negative; this can break the bell-shape property.
Expressions without the reciprocal (1 + ((x - c)/a)^b or 1 + ((x - c)/a)^(2b)) are not normalized into [0,1]; they produce values β₯ 1 (or possibly negative), so they are not valid membership degrees.
Summary: The correct bell-shaped membership function is ΞΌ(x) = 1 / (1 + ((x - c)/a)^(2b)) because it is centered at c, bounded between 0 and 1, symmetric, and its parameters have clear roles controlling center, width, and steepness.