lecture04

# lecture04 - Page 5 Simplification Example b Simplify the...

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CMPT 150 Boolean Algebra Page 1 Boolean Algebra b Boolean algebra is algebra on two-valued (binary) variables b G. Boole (1850) and C. Shannon (1938) b Operators (lowest to highest precedence) b OR b b AND b b NOT b CMPT 150 Boolean Algebra Page 2 Boolean Algebra Axioms Identity Zero Idempotence Complement Commutative Associative Distributive DeMorgan CMPT 150 Boolean Algebra Page 3 Boolean Algebra b Proving (x + y) = x y ’ using a truth table: b We can use axioms to simplify expressions b Simplification is the process of transforming a function by repeated application of axioms b This yields an equivalent new function with fewer boolean operators b Useful because each operator requires transistors CMPT 150 Boolean Algebra Page 4 Proof Using Axioms b Axioms can also be used to prove new theorems b Theorem is of the form f = g b Proof is done by using simplification on f to yield g b Example: x + xy = x

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CMPT 150 Boolean Algebra
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Unformatted text preview: Page 5 Simplification Example b Simplify the following: CMPT 150 Boolean Algebra Page 6 Logic Diagrams b A logic diagram is a graphical representation of a boolean function b I.e. a circuit of gates and wires b For example, DeMorgan’s theorem yields the following logic diagrams: b In fact, DeMorgan’s theorem provides an easy way to convert between AND’s and OR’s CMPT 150 Boolean Algebra Page 7 Logic Diagram Example b Convert the following logic diagram to all OR and NOT gates: CMPT 150 Boolean Algebra Page 8 Logic Diagram Example b Notice that a logic diagram has an equivalent representation in algebra b For example, convert the last logic diagram to algebra: b And simplify: CMPT 150 Boolean Algebra Page 9 Logic Diagram Example b We can just as easily convert algebra into a logic diagram:...
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lecture04 - Page 5 Simplification Example b Simplify the...

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