Ch2_Boolean_Algebra

Ch2_Boolean_Algebra - ECE 223 Digital Circuits and Systems...

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1 Boolean Algebra & Logic Gates ECE 223 Digital Circuits and Systems
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2 Binary (Boolean) Logic Deals with binary variables and binary logic functions Has two discrete values 0 False, Open 1 True, Close Three basic logical operations AND (.); OR (+); NOT ( )
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3 Logic Gates & Truth Tables AND OR NOT 1 0 1 0 0 1 1 0 0 A OR 1 1 0 B 1 1 0 A+B 1 0 A 1 0 0 A AND 1 1 0 B 1 0 0 A.B 0 1 A NOT AND; OR gates may have any # of inputs AND 1 if all inputs are 1; 0 other wise OR 1 if any input is 1; 0 other wise
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4 Boolean Algebra Branch of Algebra used for describing and designing two valued state variables Introduced by George Boole in 19 th centaury Shannon used it to design switching circuits (1938) Boolean Algebra Postulates An algebraic structure defined by a set of elements, B, together with two binary operators + and . that satisfy the following postulates: 1. Postulate 1: Closure with respect to both (.) and ( +) 2. Postulate 2: An identity element with respect to +, designated by 0. An identity element with respect to . designated by 1
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5 Boolean Algebra - Postulates 3. Postulate 3: Commutative with respect to + and . 4. Postulate 4: Distributive over . and + 5. Postulate 5: For each element a of B, there exist an element a such that (a) a + a = 1 and (b) a.a = 0 6. Postulate 6: There exists at least two elements a, b in B, such that a b
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6 Boolean Algebra - Postulates Postulates are facts that can be taken as true; they do not require proof We can show logic gates satisfy all the postulates 1 0 1 0 0 1 1 0 0 A OR 1 1 0 B 1 1 0 A+B 1 0 A 1 0 0 A AND 1 1 0 B 1 0 0 A.B 0 1 A NOT
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7 Boolean Algebra - Theorems Theorems help us out in manipulating Boolean expressions They must be proven from the postulates and/or other already proven theorems Exercise Prove theorems from postulates/other proven theorems
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8 Boolean Functions
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