Chapter 2

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

State De Morgan's laws for sets \(A_{i}, i \in I .\) (I is an index set.)

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 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. $$ A \cup B $$

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. $$ A^{\prime} $$

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. $$A^{\prime}$$

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. $$ A \cup B^{\prime} $$

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. $$A \cup B^{\prime}$$

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. $$ A \cap B^{\prime} $$

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. $$A \cap A^{\prime}$$

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. $$ A \oplus B $$

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. $$ A-B $$

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. $$A-B$$

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. $$ B-A $$

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. $$ A \times B $$

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. $$A \times B$$

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. $$ B \times A $$

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. $$B \times A$$

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. $$ A \times A $$

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. $$A \times A$$

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

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