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# Lec-02-Comb - C binational logic om Logic functions truth...

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CS 150 - Fall 2000 - Combinational Logic - 1 Combinational logic Logic functions, truth tables, and switches NOT, AND, OR, NAND, NOR, XOR, . . . Minimal set Axioms and theorems of Boolean algebra Proofs by re-writing Proofs by perfect induction Gate logic Networks of Boolean functions Time behavior Canonical forms Two-level Incompletely specified functions Simplification Boolean cubes and Karnaugh maps Two-level simplification

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CS 150 - Fall 2000 - Combinational Logic - 2 X Y 16 possible functions (F0–F15) 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 X and  Y X Y X or  Y not  Y not  X 1 X Y F X xor  Y X nor  Y not  (X or  Y) X = Y X nand  Y not  (X and  Y) Possible logic functions of two variables There are 16 possible functions of 2 input variables: in general, there are 2**(2**n) functions of n inputs
CS 150 - Fall 2000 - Combinational Logic - 3 Cost of different logic functions Different functions are easier or harder to implement Each has a cost associated with the number of switches needed 0 (F0) and 1 (F15): require 0 switches, directly connect output to low/high X (F3) and Y (F5): require 0 switches, output is one of inputs X' (F12) and Y' (F10): require 2 switches for "inverter" or NOT-gate X nor Y (F4) and X nand Y (F14): require 4 switches X or Y (F7) and X and Y (F1): require 6 switches X = Y (F9) and X Y (F6): require 16 switches Because NOT, NOR, and NAND are the cheapest they are the functions we implement the most in practice

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CS 150 - Fall 2000 - Combinational Logic - 4 X Y X nand Y 0 0 1 1 1 0 X Y X nor Y 0 0 1 1 1 0 X nand Y not ( (not X) nor (not Y) ) X nor Y not ( (not X) nand (not Y) ) Minimal set of functions Can we implement all logic functions from NOT, NOR, and NAND? For example, implementing X and Y is the same as implementing not (X nand Y) In fact, we can do it with only NOR or only NAND NOT is just a NAND or a NOR with both inputs tied together and NAND and NOR are "duals", i.e., easy to implement one using the other But lets not move too fast . . . lets look at the mathematical foundation of logic
CS 150 - Fall 2000 - Combinational Logic - 5 An algebraic structure An algebraic structure consists of a set of elements B binary operations { + , • } and a unary operation { ' } such that the following axioms hold: 1. set B contains at least two elements, a, b, such that a b 2. closure: a + b is in B a • b 3. commutativity: a + b = b + a a • b = b • a 4. associativity: a + (b + c) = (a + b) + c a • (b • c) = (a • b) • c 5. identity: a + 0 = a a • 1 = a 6. distributivity: a + (b • c) = (a + b) • (a + c) a • (b + c) = (a • b) + (a • c) 7. complementarity: a + a' = 1 a • a' = 0

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CS 150 - Fall 2000 - Combinational Logic - 6 Boolean algebra B = {0, 1} + is logical OR, • is logical AND ' is logical NOT All algebraic axioms hold
CS 150 - Fall 2000 - Combinational Logic - 7 X, Y are Boolean algebra variables X Y X • Y 0 0 0 0 1 0 1 0 0 1 1 1 X Y X' Y' X' • Y' ( X • Y ) + ( X' • Y' ) 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 1 ( X • Y ) + ( X' • Y' ) É X = Y X Y X' • Y 0 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 Boolean expression that is

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Lec-02-Comb - C binational logic om Logic functions truth...

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