1.(41∧A2)V(-A3) 2.( AABAD)V(A∧BA=C) The complexity of circuits depends on the complexity of proposition. We usually call the depth of proposition as delay of circuit and the number of gates as power consumption. To design a good circuit, we try to minimize delay and power consumption Example 2. Consider the boolean function majority of (A, B, C]. It means that the value of function depends on the majority of input Solution: We first consider its truth value table, we can simple connectives to represent majority m(A, B, C)=(AABAC)V(AABANC)V(AA-BAC)VGAABAC) (B∧C)V(A∧C)V(A∧B) (A∧(BVO)∨(BAC) In another way, we can depict the boolean function with a state diagram and then design the circuit accroding it. This method will be introduced in the successor course Digital Component Desing 7 Formalize Problems with Propositions Example 3. Suppose there is a murder case with three suspects. The police queried them about murder case A said, "I didn 't do it. The victim is a friend of B. And C hates him. B said, "I didn't do it. Even I don't know him. And I am not present. C said, I didn't do it. I saw A and B stayed writh the victim in that day. The murder must be one of them. Solution: Suppose only murder would lie. We try to formalize it with the following propositions A: a killed victim 2. BKV: B knows the victim 3. AP: A is present 4. CHV: C hates the victim 5.(A∧=B)V(A∧B): murder is either a or B Now we can represent the satement of each sucpects as following 1.A:-A∧BKV∧CHV 2. B: BABKVA-BP 3. C: CA APABPA((AAB)V(AAB)1. (A1 ∧ A2) ∨ (¬A3)) 2. (A ∧ B ∧ D) ∨ (A ∧ B ∧ ¬C) The complexity of circuits depends on the complexity of proposition. We usually call the depth of proposition as delay of circuit and the number of gates as power consumption. To design a good circuit, we try to minimize delay and power consumption. Example 2. Consider the boolean function majority of {A, B, C}. It means that the value of function depends on the majority of input. Solution: We first consider its truth value table, we can simple connectives to represent majority. m(A, B, C) = (A ∧ B ∧ C) ∨ (A ∧ B ∧ ¬C) ∨ (A ∧ ¬B ∧ C) ∨ (¬A ∧ B ∧ C) = (B ∧ C) ∨ (A ∧ C) ∨ (A ∧ B) = (A ∧ (B ∨ C)) ∨ (B ∧ C) In another way, we can depict the boolean function with a state diagram and then design the circuit accroding it. This method will be introduced in the successor course Digital Component Desing. 7 Formalize Problems with Propositions Example 3. Suppose there is a murder case with three suspects. The police queried them about murder case. A said,”I didn’t do it. The victim is a friend of B. And C hates him.” B said, ”I didn’t do it. Even I don’t know him. And I am not present.” C said, ”I didn’t do it. I saw A and B stayed with the victim in that day. The murder must be one of them.” Solution: Suppose only murder would lie. We try to formalize it with the following propositions: 1. A: A killed victim. 2. BKV : B knows the victim. 3. AP: A is present. 4. CHV : C hates the victim. 5. (A ∧ ¬B) ∨ (¬A ∧ B): murder is either A or B. Now we can represent the satement of each sucpects as following: 1. A: ¬A ∧ BKV ∧ CHV . 2. B: ¬B ∧ ¬BKV ∧ ¬BP. 3. C: ¬C ∧ AP ∧ BP ∧ ((A ∧ ¬B) ∨ (¬A ∧ B)). 4