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Compute the value of adding the following two fuzzy integers:

A={(0.3,1), (0.6,2), (1,3), (0.7,4), (0.2,5)}

B={(0.5,11), (1, 12), (0.5, 13)}

Where fuzzy addition is defined as

$\mu_{A+B} (z) = max_{x+y=z} (min (\mu_A(x), \mu_b(x)))$

Then, f(A+B) is equal to

  1. {(0.5, 12), (0.6, 13), (1, 14), (0.7, 15), (0.7, 16), (1, 17), (1, 18)}
  2. {(0.5, 12), (0.6, 13), (1, 14), (1, 15), (1, 16), (1, 17), (1, 18)}
  3. {(0.3, 12), (0.5, 13), (0.5, 14), (1, 15), (0.7, 16), (0.5, 17), (0.2, 18)}
  4. {(0.3, 12), (0.5, 13), (0.6, 14), (1, 15), (0.7, 16), (0.5, 17), (0.2, 18)}
in Others by Veteran (105k points)
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1 Answer

+5 votes
Best answer

μA+B(z)=max x+y=z(min(μA(x),μB(x)))

A={(0.3,1), (0.6,2), (1,3), (0.7,4), (0.2,5)}

B={(0.5,11), (1, 12), (0.5, 13)}

first add the numbers(x+y=z) and write the min membership value since function is min((μA(x),μB(x)) u will get follwing 15 terms

{(0.3,12),(0.3,13),(0.3,14),(0.5,13),(0.6,14),(0.5,15),(0.5,14),(1,15),(0.5,16),(0.5,15),(0.7,16),(0.5,17),(0.2,16),(0.2,17),(0.2,18)}

now write all distinct elements with max membership (written in bold)  value since max is there in the question

ANS WILL BE D     {(0.3, 12), (0.5, 13), (0.6, 14), (1, 15), (0.7, 16), (0.5, 17), (0.2, 18)}

by Boss (48.8k points)
selected by
0
Please explain this step
(0.3,12),(0.3,13),(0.3,14) you choose (0.3,12)
And here (0.5,14),(1,15),(0.5,16) you  (1,15)
And here you said max again u had choose (0.5,17)
Why please explain it
0
see there is no other instance of 12 other than (0.3,12) so it is directly taken  as it is and

it is not chosen among (0.3,12),(0.3,13),(0.3,14)

and in the case of 15 there are 3 instances with membership .5,1, .5  and among these max i.e 1 is taken

and so on
0
Thanks a lot for explanation
+1
Hello Sanjay ..can u explain it properly specially second half
+1

It has been taken like,

with 12, there is only (0.3, 12)
with 13, there are (0.3,13) (0.5,13). thus max is (0.5,13)

with 14, there are (0.3,14) (0.6,14) (0.5,14). thus max is (0.6,14)

this continues for all the terms. 

Last we will get,
{(0.3, 12), (0.5, 13), (0.6, 14), (1, 15), (0.7, 16), (0.5, 17), (0.2, 18)}

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