Suppose the function y and a fuzzy integer number around - 4 for x are given…
2015
Suppose the function y and a fuzzy integer number around - 4 for x are given as \(y=(x-3)^2 + 2\).
Around - 4 ={(2, 0.3), (3, 0.6), (4, 1), (5, 0.6), (6, 0.3)} respectively. Then f(Around-4) is given by :
- A.
{(2, 0.6), (3, 0.3), (6, 1), (11, 0.3)}
- B.
{(2, 0.6), (3, 1), (6, 1), (11, 0.3)}
- C.
{(2, 0.6), (3, 1), (6, 0.6), (11, 0.3)}
- D.
{(2, 0.6), (3, 0.3), (6, 0.6), (11, 0.3)}
Attempted by 36 students.
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Correct answer: C
Solution:
Step 1: Apply f(x) = (x - 3)^2 + 2 to each element of the fuzzy set for x and keep their memberships.
x = 2 maps to f(2) = (2-3)^2 + 2 = 3 with membership 0.3.
x = 3 maps to f(3) = (3-3)^2 + 2 = 2 with membership 0.6.
x = 4 maps to f(4) = (4-3)^2 + 2 = 3 with membership 1.
x = 5 maps to f(5) = (5-3)^2 + 2 = 6 with membership 0.6.
x = 6 maps to f(6) = (6-3)^2 + 2 = 11 with membership 0.3.
Step 2: If multiple x values map to the same image y, take the maximum membership among their preimages (the supremum).
Image 2 comes from x = 3, so membership = 0.6.
Image 3 comes from x = 2 (0.3) and x = 4 (1); take max ⇒ membership = 1.
Image 6 comes from x = 5, so membership = 0.6.
Image 11 comes from x = 6, so membership = 0.3.
Final answer: f(Around-4) = {(2, 0.6), (3, 1), (6, 0.6), (11, 0.3)}.