Computer Science – 15.2 Boolean Algebra and Logic Circuits | e-Consult
15.2 Boolean Algebra and Logic Circuits (1 questions)
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Answer:
- The expression is (A → B) ∧ (¬B → ¬A).
- We can use the implication equivalence: A → B ≡ ¬A ∨ B. Therefore, A → B becomes ¬A ∨ B. Similarly, ¬B → ¬A becomes B ∨ ¬A.
- Substituting these into the original expression, we get: (¬A ∨ B) ∧ (B ∨ ¬A).
- Using the distributive law: (¬A ∨ B) ∧ (B ∨ ¬A) ≡ (¬A ∧ B ∨ ¬A ∧ ¬A) ∨ (B ∧ B ∨ B ∧ ¬A).
- Simplifying the conjunctions: (¬A ∧ B ∨ False) ∨ (B ∨ B ∧ ¬A).
- This simplifies to: (¬A ∧ B) ∨ (True) ∨ (B ∧ ¬A).
- Since True ≡ True, the expression becomes: (¬A ∧ B) ∨ True ∨ (B ∧ ¬A).
- The OR operation with True is idempotent, so the expression simplifies to: True.
- Therefore, the simplified expression is True.