By Joachim Stolze, Dieter Suter

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**Additional info for Quantum Computing: A Short Course from Theory to Experiment**

**Sample text**

This is ∼ ∼ the membership mapping and is shown in Fig. 7. 1 Fuzzy Set Operations Considering three fuzzy sets A, B and C on the universe X. For a given ∼ ∼ ∼ element x of the universe, the following function theoretic operations for the set theoretic operations unions, intersection and complement are deﬁned for A, B and C on X: ∼ ∼ Union: ∼ µA ∪ B (x) = µA (x)VµB (x). ∼ ∼ ∼ ∼ Intersection: µA ∩ B (x) = µA (x)ΛµB (x). ∼ ∼ ∼ ∼ Complement µ− (x) = 1 − µA (x). 3 Fuzzy Sets 21 mx(x) A ⊆ X → mA (x) ~ for all x ∈ X mf (x) = 0.

7. 8 Solution. 9 ⎦ . 5 Tolerance and Equivalence Relations Relations exhibit various other properties apart from that discussed in Sects. 4. It is already said that the relation can be used in graph theory. The various other properties that are dealt here include reﬂexivity, symmetry, and transitivity. These are discussed in detail for the crisp and fuzzy relations and are called as equivalence relation. Apart from these, tolerance relations of both fuzzy and crisp relations are also described.

2) LHS = RHS A ∪ (B ∪ C) = (A ∪ B) ∪ C. 2) 18 2 Classical Sets and Fuzzy Sets 2. (A ∩ (B ∩ C) = (A ∩ B) ∩ C LHS (a) (B ∩ C) = {1}. (b) A ∩ (B ∩ C) = {φ}. 3) RHS (A ∩ B) ∩ C (a) (A ∩ B) = {9}. (b) (A ∩ B) ∩ C = {φ}. 4) LHS = RHS A ∩ (B ∩ C) = (A ∩ B) ∩ C. Thus associative property is proved. The distributive property is given by, 1. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) LHS (a) B ∩ C = {1}. (b) A ∪ (B ∩ C) = {9, 5, 6, 8, 10, 1}. 5) RHS (A ∪ B) ∩ (A ∪ C) (a) (A ∪ B) = {9, 5, 6, 8, 10, 1, 2, 3, 7}. (b) (A ∪ C) = {9, 5, 6, 8, 10, 1, 0}.

### Quantum Computing: A Short Course from Theory to Experiment by Joachim Stolze, Dieter Suter

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