Section 10 2 Boolean Algebra Motivation Notice the
Section 10. 2 Boolean Algebra Motivation: Notice the list of corresponding properties for the algebra of sets and the algebra of propositional wffs. Propositional Wffs; false, true, , , ¬ Properties: A B B A A false A (A B) C A (B C) A B B A A true A (A B) C A (B C) (A B) (A C) A ¬ A false A ¬ A true. power(S); , S, , , ' Properties: A B=B A A =A (A B) C = A (B C) A B=B A A S=A (A B) C = A (B C) = (A B) (A C) A A'= A A ' = S. These two algebras are concrete examples of a Boolean algebra which has the following properties: 1. + and · are commutative and associative with identity elements 0 and 1, respectively. 2. + and · distribute over each other. 1
Properties of Boolean algebra operations. For any property of the operations there is a dual property obtained by interchanging 0 with 1, and + with ·. Similarly, any proof has a dual proof obtained in the same way. Here are some basic properties of the operations. Quiz (2 minutes). Simplify the expression Answer. (commute and associate) (absorption) (associate and absorption) 2
Digital Circuits A digital circuit (or logic circuit) is an electronic representation of a truth function. The following three logic gates can be used to implement any digital circuit. x y xy AND gate x y x+y OR gate Quiz. Simplify the following digital circuit. x NOT gate (inverter) x y Solution. The circuit represents the following wff. (distribute) (absorption, absorption) (absorption) So the circuit can be implemented by a single AND gate. 3
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