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PART 2 Fuzzy sets vs crisp sets 1. Properties of -cuts 2. Fuzzy set representations 3. Extension principle FUZZY SETS AND FUZZY LOGIC Theory and Applications

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Properties of -cuts Theorem 2.1 Let A, B F (X). Then, the following properties hold for all , [0, 1]: (i) (ii) (iii) (iv) (v)

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Properties of -cuts

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Theorem 2.2 Let A i F (X) for all i I, where I is an index set. Then, (vi) (vii)

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Properties of -cuts

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Properties of -cuts Theorem 2.3 Let A, B F (X). Then, for all [0, 1], (viii) (ix)

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Properties of -cuts Theorem 2.4 For any A F (X), the following properties hold: (x) (xi)

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Fuzzy set representations Theorem 2.5 (First Decomposition Theorem) For every A F (X), where A is defined by (2.1), and denotes the standard fuzzy union.

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Fuzzy set representations

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Theorem 2.6 (Second Decomposition Theorem) For every A F (X), where + A denotes a special fuzzy set defined by and denotes the standard fuzzy union.

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Fuzzy set representations Theorem 2.7 (Third Decomposition Theorem) For every A F (X), where (A) is the level of A, A is defined by (2.1), and denotes the standard fuzzy union.

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Fuzzy set representations

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Extension principle

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Extension principle. Any given function f : XY induces two functions,

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Extension principle which are defined by for all A F (X) and for all B F (Y).

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Extension principle

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Theorem 2.8 Let f : XY be an arbitrary crisp function. Then, for any A i F (X) and any B i F (Y), i I, the following properties of functions obtained by the extension principle hold:

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Extension principle

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Theorem 2.9 Let f : XY be an arbitrary crisp function. Then, for any A i F (X) and all [0, 1] the following properties of fuzzified by the extension principle hold:

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Extension principle

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Extension principle

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Extension principle Theorem 2.10 Let f : XY be an arbitrary crisp function. Then, for any A i F (X), f fuzzified by the extension principle satisfies the equation:

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Exercise 2 2.4 2.8 2.11