91Ó°ÊÓ

Arrange the binary words of the given length in increasing order of magnitude. Length two.

The empty set is a subset of every set. (Hint: Consider the implication \(x \in \emptyset \rightarrow x \in A .\) )

The empty set is unique. (Hint: Assume there are two empty sets, \(\emptyset_{1}\) and \(\emptyset_{2}\). Then use Exercise 53.)

Simplify each set expression. $$ (A \cap B)^{\prime} \cup\left(A \cup B^{\prime}\right) $$

Simplify each set expression. $$\left(A^{\prime} \cup B^{\prime}\right)^{\prime} \cup\left(A^{\prime} \cap B\right)$$

State the distributive laws using the sets \(A\) and \(B_{i}, i \in I\)

The sum of two fuzzy sets \(A\) and \(B\) is the fuzzy \(\operatorname{set} A \oplus B,\) where \(d_{A}+B(x)=\) \(1 \wedge\left|d_{A}(x)+d_{B}(x)\right| ;\) their difference is the fuzzy \(\operatorname{set} A-B,\) where \(d_{A-B}(x)=\) \(0 \vee\left|d_{A}(x)-d_{B}(x)\right| ;\) and their cartesian product is the fuzzy set \(A \times B\) where \(d_{A \times B}(x, y)=d_{A}(x) \wedge d_{B}(x) .\) Use the fuzzy sets \(A=\\{\text { Angelo } 0.4, \text { Bart } 0.7, \text { Cathy } 0.6\\}\) and \(B=\\{\text { Dan } 0.3, \text { Elsie } 0.8, \text { Frank } 0.4\\}\) to find each fuzzy set. $$B-A$$

The sum of two fuzzy sets \(A\) and \(B\) is the fuzzy set \(A \oplus B,\) where \(d_{A \oplus B}(x)=\) 1\(\wedge\left|d_{A}(x)+d_{B}(x)\right|\) itheir difference is the fuzzy set \(A-B,\) where \(d_{A-B}(x)=\) \(0 \vee\left[d_{A}(x)-d_{B}(x)\right] ;\) and their eartesian produet is the fuzzy set \(A \times B\) where \(d_{A \times B}(x, y)=d_{A}(x) \wedge d_{B}(x) .\) Use the fuzzy sets \(A=\\{\text { Angelo } 0.4, \text { Bart }\) \(0.7,\) Cathy 0.6\(\\}\) and \(B=\\{\operatorname{Dan} 0.3, \text { Elsie } 0.8, \text { Frank } 0.4\\}\) to find each fuzzy set. $$ A \times A $$

Let \(A\) and \(B\) be any fuzzy sets. Prove each. $$(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}$$