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Digital Logic Circuits – Chapter 1 Section 1-3, 1-2

Boolean Algebra What is “Algebra”? –A branch of mathematics in which symbols represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set.

Boolean Algebra What is “Boolean”? –Of or relating to a data type or variable in a programming language that can have one of two values, true or false.

Boolean Algebra Therefore –Boolean algebra is an algebra that deals with binary variables and logic operations But why is it called “Boolean”? –Invented by English mathematician George Boole in the 1850’s (published in 1854)

Boolean Algebra Boolean algebra consists of… –A set of symbols that represent variables Use letters just like “regular algebra” A, B, C, a, b, c… –Variables are binary (2-valued) 0, 1 true, false –Three basic operators AND, OR, NOT –Other symbols ( ) =

Boolean Operators AND –Notation: A · B, AB, (AB), A(B) –Yields a value of 1 when both A and B are 1 –Yields a value of 0 when either A or B is 0

Boolean Operators OR –Notation: A + B –Yields a value of 1 when either A or B is 1 –Yields a value of 0 when both A and B are 0

Boolean Operators NOT –Notation: A, A’ –Yields a value of 1 A is 0 –Yields a value of 0 when A is 1

Boolean Expressions As in “regular” algebra, variables, operators, and symbols can be combined to form expressions or functions F(x, a, b) = x + (a · b) –F is a boolean function of three variables –Often written as: F = x + (a · b)

Boolean Functions Solution is found by plugging in values for the variables and applying the operators F = x + (a · b) –Given x=0, a=1, b=0 we have F(0, 1, 0) = 0 + (1 · 0) F(0, 1, 0) = 0 + (1 · 1) F(0, 1, 0) = 0 + 1 F(0, 1, 0) = 1

Operator Precedence NOT operators are done first Parenthesized expressions second AND operators are done third (left to right) OR operators are done fourth (left to right) Thus AB + CD is the same as (AB) + (CD), not A(B + C)D

Boolean Functions Typically, we want to exhaustively evaluate a given Boolean function –That is, we want to know its functional value for every possible combination of inputs –This leads us to Truth Tables

Truth Tables List all possible combinations of input values in the left hand columns List expression result in the right hand column ABAB 000 010 100 111 ABA+B 000 011 101 111 AA 01 10

Truth Tables For more complex expressions you will [typically] create columns for each operation: F(x, a, b) = x + (a · b) xabb(a·b)x+(a·b ) 000100 001000 010111 011000 100101 101001 110111 111001

Truth Tables Doesn’t matter what order the input variables are, just a long as all combinations are included. xabb(a·b)x+(a·b) 000100 100101 001000 101001 010111 110111 011000 111001

Truth Tables Are they the same? xabb(a·b)x+(a·b) 000100 100101 001000 101001 010111 110111 011000 111001 xabb(a·b)x+(a·b) 000100 001000 010111 011000 100101 101001 110111 111001

Proof via Truth Table Proof that our earlier conjecture (AB) + (CD) ≠ A(B + C)D is true via truth table

Other Methods Truth tables can always be used to evaluate Boolean expressions But, they’re not always convenient –They can get big and unmanageable quite fast Fortunately, as in any good algebra, Boole gave us a set of axioms (laws, identities) that can be used to simplify boolean expressions symbolically

Axioms 1.x + 0 = x 2.x + 1 = 1 3.x + x = x 4.x + x’ = 1 5.x + y = y + x 6.x + (y + z) = (x + y) + z 7.x(y + z) = xy + xz 8.(x + y)’ = x’y’ 9.(x’)’ = x 10.x · 1 = x 11.x · 0 = 0 12.x · x = x 13.x · x’ = 0 14.xy = yx 15.x(yz) = (xy)z 16.x + yz = (x + y)(x + z) 17.(xy)’ = x’ + y’

Axioms 1, 10 – identity laws 2, 11 – one, zero laws 3, 12 – idempotence laws 4, 13 – inverse laws 5, 14 – commutative laws 6, 15 – associative laws 7, 16 – distributive laws 8, 17 – De’ Mogan’s laws 9 – double negation law (?? – I made that name up)

Why? What’s all this stuff used for? –“The purpose of Boolean algebra is to facilitate the analysis and design of digital circuits.” Express a truth table relationship in algebraic (symbolic) notation Express a logic diagram’s input/output relationships in algebraic notation Provides a means for finding simpler circuits to implement a given function

Logic Circuits These are the things computers (and other digital devices) are made of Circuit designers use Boolean algebra to design circuits drawn on schematic drawings Fabrication facilities use schematic drawings to produce silicon chips AND OR NOT Schematic Symbols

Logic Circuits F(x, a, b) = x + (a · b) AND OR NOT b a x AND OR b a x b a x

Homework Prove all 17 axioms using truth tables Due next lecture

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