For the fuzzy sets A and B, the membership vectors are μA(x) = {0.2, 0.4, 0.8,…

2012

For the fuzzy sets A and B, the membership vectors are μA(x) = {0.2, 0.4, 0.8, 0.5, 0.1} and μB(x) = {0.1, 0.3, 0.6, 0.3, 0.2}. Under the standard minimum operator for fuzzy intersection, what is μA∩B(x)?

Answer: C. {0.1, 0.3, 0.6, 0.3, 0.1} — ConceptFor two fuzzy sets on the same universe, the standard intersection uses the minimum t-norm. At each universe element x, μA∩B(x) = min(μA(x), μB(x));…

  1. A.

    {0.9, 0.7, 0.4, 0.8, 0.9}

  2. B.

    {0.2, 0.4, 0.8, 0.5, 0.2}

  3. C.

    {0.1, 0.3, 0.6, 0.3, 0.1}

  4. D.

    {0.7, 0.3, 0.4, 0.2, 0.7}

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Correct answer: C

Concept

For two fuzzy sets on the same universe, the standard intersection uses the minimum t-norm.

At each universe element x, μA∩B(x) = min(μA(x), μB(x)); each position is evaluated independently.

Application

  1. First element: min(0.2, 0.1) = 0.1.

  2. Second element: min(0.4, 0.3) = 0.3.

  3. Third element: min(0.8, 0.6) = 0.6.

  4. Fourth element: min(0.5, 0.3) = 0.3.

  5. Fifth element: min(0.1, 0.2) = 0.1.

Cross-check

Every resulting grade is no greater than the corresponding grade in either input set, and each equals one of the two grades at that position, as a componentwise minimum must.

Therefore, μA∩B(x) = {0.1, 0.3, 0.6, 0.3, 0.1}.

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