04-Kmap - CS231 Boolean Algebra 1 Minter ms, Maxter ms, and...

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Unformatted text preview: CS231 Boolean Algebra 1 Minter ms, Maxter ms, and K-Maps CS231 Boolean Algebra 2 Summar y so far So far: A bunch of Boolean algebra trickery for simplifying expressions and circuits The algebra guarantees us that the simplified circuit is equivalent to the original one Next: An alternative simplification method Well start using all this stuff to build and analyze bigger, more useful, circuits CS231 Boolean Algebra 3 Standar d for ms of expr essions We can write expressions in many ways, but some ways are more useful than others A sum of products (SOP) expression contains: Only OR (sum) operations at the outermost level Each term that is summed must be a product of literals The advantage is that any sum of products expression can be implemented using a two-level circuit literals and their complements at the 0th level AND gates at the first level a single OR gate at the second level This diagram uses some shorthands NOT gates are implicit literals are reused this is not okay in LogicWorks! f(x,y,z) = y + xyz + xz CS231 Boolean Algebra 4 Minter ms A minterm is a special product of literals, in which each input variable appears exactly once. A function with n variables has 2 n minterms (since each variable can appear complemented or not) A three-variable function, such as f(x,y,z), has 2 3 = 8 minterms: Each minterm is true for exactly one combination of inputs: xyz xyz xyz xyz xyz xyz xyz xyz Minterm I s true when Shorthand xyz x=0, y=0, z=0 m xyz x=0, y=0, z=1 m 1 xyz x=0, y=1, z=0 m 2 xyz x=0, y=1, z=1 m 3 xyz x=1, y=0, z=0 m 4 xyz x=1, y=0, z=1 m 5 xyz x=1, y=1, z=0 m 6 xyz x=1, y=1, z=1 m 7 CS231 Boolean Algebra 5 Sum of minter ms for m Every function can be written as a sum of minterms , which is a special kind of sum of products form The sum of minterms form for any function is unique I f you have a truth table for a function, you can write a sum of minterms expression just by picking out the rows of the table where the function output is 1. x y z f (x,y,z) f (x,y,z) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 f = xyz + xyz + xyz + xyz + xyz = m + m 1 + m 2 + m 3 + m 6 = m(0,1,2,3,6) f = xyz + xyz + xyz = m 4 + m 5 + m 7 = m(4,5,7) f contains all the minterms not in f CS231 Boolean Algebra 6 The dual idea: pr oducts of sums Just to keep you on your toes... A product of sums (POS) expression contains: Only AND (product) operations at the outermost level Each term must be a sum of literals Product of sums expressions can be implemented with two-level circuits literals and their complements at the 0th level OR gates at the first level a single AND gate at the second level Compare this with sums of products f(x,y,z) = y (x + y + z) (x + z) CS231 Boolean Algebra...
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This note was uploaded on 04/15/2008 for the course CS 231 taught by Professor - during the Spring '08 term at University of Illinois at Urbana–Champaign.

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04-Kmap - CS231 Boolean Algebra 1 Minter ms, Maxter ms, and...

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