Charles Kime & Thomas Kaminski © 2008 Pearson Education, Inc. (Edited by Dr. Muhamed Mudawar for COE 202 & EE 200 at KFUPM) Boolean Algebra and Logic Gates Logic and Computer Design Fundamentals

Boolean Algebra and Logic Gates 2 Binary Logic and Gates  Binary variables take on one of two values.  Logical operators operate on binary values and binary variables.  Basic logical operators are the logic functions AND, OR and NOT.  Logic gates implement logic functions.  Boolean Algebra: a useful mathematical system for specifying and transforming logic functions.  We study Boolean algebra as a foundation for designing and analyzing digital systems!

Boolean Algebra and Logic Gates 3 Binary Variables  Recall that the two binary values have different names: True/False On/Off Yes/No 1/0  We use 1 and 0 to denote the two values.  Variable identifier examples: A, B, y, z, or X 1 for now RESET, START_IT, or ADD1 later

Boolean Algebra and Logic Gates 4 Logical Operations  The three basic logical operations are: AND OR NOT  AND is denoted by a dot (·).  OR is denoted by a plus (+).  NOT is denoted by an overbar ( ¯ ), a single quote mark (') after, or (~) before the variable.

Boolean Algebra and Logic Gates 5  Examples: is read “Y is equal to A AND B.” is read “z is equal to x OR y.” is read “X is equal to NOT A.” Notation Examples  Note: The statement: = 2 (read “one plus one equals two”) is not the same as = 1 (read “1 or 1 equals 1”).  BAY  yxz   AX 

Boolean Algebra and Logic Gates 6 Operator Definitions  Operations are defined on the values "0" and "1" for each operator: AND 0 · 0 = 0 0 · 1 = 0 1 · 0 = 0 1 · 1 = 1 OR = = = = 1 NOT 10  01 

Boolean Algebra and Logic Gates X NOT XZ  Truth Tables  Tabular listing of the values of a function for all possible combinations of values on its arguments  Example: Truth tables for the basic logic operations: Z = X·Y YX AND OR XYZ = X+Y

Boolean Algebra and Logic Gates 8 Truth Tables – Cont’d  Used to evaluate any logic function  Consider F(X, Y, Z) = X Y + Y Z XYZX YYY ZF = X Y + Y Z

Boolean Algebra and Logic Gates 9  Using Switches Inputs:  logic 1 is switch closed  logic 0 is switch open Outputs:  logic 1 is light on  logic 0 is light off. NOT input:  logic 1 is switch open  logic 0 is switch closed Logic Function Implementation Switches in series => AND Switches in parallel => OR C Normally-closed switch => NOT

Boolean Algebra and Logic Gates 10  Example: Logic Using Switches  Light is on (L = 1) for L(A, B, C, D) = and off (L = 0), otherwise.  Useful model for relay and CMOS gate circuits, the foundation of current digital logic circuits Logic Function Implementation – cont’d B A D C A (B C + D) = A B C + A D

Boolean Algebra and Logic Gates 11 Logic Gates  In the earliest computers, switches were opened and closed by magnetic fields produced by energizing coils in relays. The switches in turn opened and closed the current paths.  Later, vacuum tubes that open and close current paths electronically replaced relays.  Today, transistors are used as electronic switches that open and close current paths.

Boolean Algebra and Logic Gates 12 Logic Gate Symbols and Behavior  Logic gates have special symbols:  And waveform behavior in time as follows : X 0011 Y0101 X · Y(AND)0001 X+ Y(OR)0111 (NOT)X 1100 OR gate X Y Z= X+ Y X Y Z= X · Y AND gate X Z= X NOT gate or inverter

Boolean Algebra and Logic Gates 13 Logic Diagrams and Expressions  Boolean equations, truth tables and logic diagrams describe the same function!  Truth tables are unique, but expressions and logic diagrams are not. This gives flexibility in implementing functions. X Y F Z Logic Diagram Logic Equation ZY X F  Truth Table X Y Z Z Y X F   

Boolean Algebra and Logic Gates 14 Gate Delay  In actual physical gates, if an input changes that causes the output to change, the output change does not occur instantaneously.  The delay between an input change and the output change is the gate delay denoted by t G : tGtG tGtG Input Output Time (ns) t G = 0.3 ns

Boolean Algebra and Logic Gates Commutative Associative Distributive DeMorgan’s X. 1 X = X. 00 = X. XX = 0 = Boolean Algebra X + YY + X = (X + Y)Z + X + (YZ)Z) += X(Y + Z)XYXZ += X + YX. Y = XYYX = (XY)ZX(YX(YZ)Z) = X+ YZ(X + Y)(X + Z)= X. YX + Y = X + 0 X = + X 11 = X + XX = 1 = X = X  Invented by George Boole in 1854  An algebraic structure defined by a set B = {0, 1}, together with two binary operators (+ and ·) and a unary operator ( ) Idempotence Complement Involution Identity element

Boolean Algebra and Logic Gates 16 Some Properties of Boolean Algebra  Boolean Algebra is defined in general by a set B that can have more than two values  A two-valued Boolean algebra is also know as Switching Algebra. The Boolean set B is restricted to 0 and 1. Switching circuits can be represented by this algebra.  The dual of an algebraic expression is obtained by interchanging + and · and interchanging 0’s and 1’s.  The identities appear in dual pairs. When there is only one identity on a line the identity is self-dual, i. e., the dual expression = the original expression.  Sometimes, the dot symbol ‘  ’ (AND operator) is not written when the meaning is clear

Boolean Algebra and Logic Gates 17  Example: F = (A + C) · B + 0 dual F = (A · C + B) · 1 = A · C + B  Example: G = X · Y + (W + Z) dual G =  Example: H = A · B + A · C + B · C dual H =  Unless it happens to be self-dual, the dual of an expression does not equal the expression itself  Are any of these functions self-dual? (A+B)(A+C)(B+C)=(A+BC)(B+C)=AB+AC+BC Dual of a Boolean Expression (X+Y) · (W · Z) = (X+Y) · (W+Z) (A+B) · (A+C) · (B+C) H is self-dual

Boolean Algebra and Logic Gates 18 Boolean Operator Precedence  The order of evaluation is: 1.Parentheses 2.NOT 3.AND 4.OR  Consequence: Parentheses appear around OR expressions  Example: F = A(B + C)(C + D)

Boolean Algebra and Logic Gates 19 Boolean Algebraic Proof – Example 1  A + A · B = A (Absorption Theorem) Proof StepsJustification A + A · B = A · 1 + A · B Identity element: A · 1 = A = A · ( 1 + B) Distributive = A · B = 1 = A Identity element  Our primary reason for doing proofs is to learn: Careful and efficient use of the identities and theorems of Boolean algebra, and How to choose the appropriate identity or theorem to apply to make forward progress, irrespective of the application.

Boolean Algebra and Logic Gates 20  AB + AC + BC = AB + AC (Consensus Theorem) Proof StepsJustification = AB + AC + BC = AB + AC + 1 · BCIdentity element = AB + AC + (A + A) · BCComplement = AB + AC + ABC + ABCDistributive = AB + ABC + AC + ACBCommutative = AB · 1 + ABC + AC · 1 + ACBIdentity element = AB (1+C) + AC (1 + B)Distributive = AB. 1 + AC. 11+X = 1 = AB + ACIdentity element Boolean Algebraic Proof – Example 2

Boolean Algebra and Logic Gates 21 Useful Theorems  Minimization X Y + X Y = Y  Absorption X + X Y = X  Simplification X + X Y = X + Y  DeMorgan’s  X + Y = X · Y  Minimization (dual) (X+Y)(X+Y) = Y  Absorption (dual) X · (X + Y) = X  Simplification (dual) X · (X + Y) = X · Y  DeMorgan’s (dual)  X · Y = X + Y

Boolean Algebra and Logic Gates 22 Truth Table to Verify DeMorgan’s XYX·YX·YX+YXY X · YX·YX·YX+Y X + Y = X · Y X · Y = X + Y  Generalized DeMorgan’s Theorem: X 1 + X 2 + … + X n = X 1 · X 2 · … · X n X 1 · X 2 · … · X n = X 1 + X 2 + … + X n

Boolean Algebra and Logic Gates 23 Complementing Functions  Use DeMorgan's Theorem: 1.Interchange AND and OR operators 2.Complement each constant and literal  Example: Complement F = F = (x + y + z)(x + y + z)  Example: Complement G = (a + bc)d + e G = (a (b + c) + d) e x  zyzyx

Boolean Algebra and Logic Gates 24 Expression Simplification  An application of Boolean algebra  Simplify to contain the smallest number of literals (variables that may or may not be complemented) = AB + ABCD + A C D + A C D + A B D = AB + AB(CD) + A C (D + D) + A B D = AB + A C + A B D = B(A + AD) +AC = B (A + D) + A C (has only 5 literals)  DCBADCADBADCABA

Boolean Algebra and Logic Gates 25 Next … Canonical Forms  Minterms and Maxterms  Sum-of-Minterm (SOM) Canonical Form  Product-of-Maxterm (POM) Canonical Form  Representation of Complements of Functions  Conversions between Representations

Boolean Algebra and Logic Gates 26 Minterms  Minterms are AND terms with every variable present in either true or complemented form.  Given that each binary variable may appear normal (e.g., x) or complemented (e.g., ), there are 2 n minterms for n variables.  Example: Two variables (X and Y) produce 2 x 2 = 4 combinations: (both normal) (X normal, Y complemented) (X complemented, Y normal) (both complemented)  Thus there are four minterms of two variables. YX XY YX YX x

Boolean Algebra and Logic Gates 27 Maxterms  Maxterms are OR terms with every variable in true or complemented form.  Given that each binary variable may appear normal (e.g., x) or complemented (e.g., x), there are 2 n maxterms for n variables.  Example: Two variables (X and Y) produce 2 x 2 = 4 combinations: (both normal) (x normal, y complemented) (x complemented, y normal) (both complemented) YX  YX  YX  YX 

Boolean Algebra and Logic Gates 28  Two variable minterms and maxterms.  The minterm m i should evaluate to 1 for each combination of x and y.  The maxterm is the complement of the minterm Minterms & Maxterms for 2 variables xyIndexMintermMaxterm 000m 0 = x yM 0 = x + y 011m 1 = x yM 1 = x + y 102m 2 = x yM 2 = x + y 113m 3 = x yM 3 = x + y

Boolean Algebra and Logic Gates 29 Minterms & Maxterms for 3 variables M 3 = x + y + zm 3 = x y z3110 M 4 = x + y + zm 4 = x y z4001 M 5 = x + y + zm 5 = x y z5101 M 6 = x + y + zm 6 = x y z y x z M 7 = x + y + zm 7 = x y z7 M 2 = x + y + zm 2 = x y z2 M 1 = x + y + zm 1 = x y z1 M 0 = x + y + zm 0 = x y z0 MaxtermMintermIndex Maxterm M i is the complement of minterm m i M i = m i and m i = M i

Boolean Algebra and Logic Gates 30 Purpose of the Index  Minterms and Maxterms are designated with an index  The index number corresponds to a binary pattern  The index for the minterm or maxterm, expressed as a binary number, is used to determine whether the variable is shown in the true or complemented form  For Minterms: ‘1’ means the variable is “Not Complemented” and ‘0’ means the variable is “Complemented”.  For Maxterms: ‘0’ means the variable is “Not Complemented” and ‘1’ means the variable is “Complemented”.

Boolean Algebra and Logic Gates 31 Standard Order  All variables should be present in a minterm or maxterm and should be listed in the same order (usually alphabetically)  Example: For variables a, b, c: Maxterms (a + b + c), (a + b + c) are in standard order However, (b + a + c) is NOT in standard order (a + c) does NOT contain all variables Minterms (a b c) and (a b c) are in standard order However, (b a c) is not in standard order (a c) does not contain all variables

Boolean Algebra and Logic Gates 32 Sum-Of-Minterm (SOM)  Sum-Of-Minterm (SOM) canonical form: Sum of minterms of entries that evaluate to ‘1’ xyzFMinterm m 1 = x y z m 6 = x y z 1111m 7 = x y z F = m 1 + m 6 + m 7 = ∑ (1, 6, 7) = x y z + x y z + x y z Focus on the ‘1’ entries

Boolean Algebra and Logic Gates 33 + a b c d Sum-Of-Minterm Examples  F(a, b, c, d) = ∑(2, 3, 6, 10, 11)  F(a, b, c, d) = m 2 + m 3 + m 6 + m 10 + m 11  G(a, b, c, d) = ∑(0, 1, 12, 15)  G(a, b, c, d) = m 0 + m 1 + m 12 + m 15 + a b c da b c d+ a b c d a b c d+ a b c d

Boolean Algebra and Logic Gates 34 Product-Of-Maxterm (POM)  Product-Of-Maxterm (POM) canonical form: Product of maxterms of entries that evaluate to ‘0’ xyzFMaxterm M 2 = (x + y + z) M 4 = (x + y + z) M 6 = (x + y + z) 1111 Focus on the ‘0’ entries F = M 2 ·M 4 ·M 6 = ∏ (2, 4, 6) = (x+y+z) (x+y+z) (x+y+z)

Boolean Algebra and Logic Gates 35  F(a, b, c, d) = ∏(1, 3, 6, 11)  F(a, b, c, d) = M 1 · M 3 · M 6 · M 11  G(a, b, c, d) = ∏(0, 4, 12, 15)  G(a, b, c, d) = M 0 · M 4 · M 12 · M 15 Product-Of-Maxterm Examples (a+b+c+d)

Boolean Algebra and Logic Gates 36 Observations  We can implement any function by "ORing" the minterms corresponding to the ‘1’ entries in the function table. A minterm evaluates to ‘1’ for its corresponding entry.  We can implement any function by "ANDing" the maxterms corresponding to ‘0’ entries in the function table. A maxterm evaluates to ‘0’ for its corresponding entry.  The same Boolean function can be expressed in two canonical ways: Sum-of-Minterms (SOM) and Product-of- Maxterms (POM).  If a Boolean function has fewer ‘1’ entries then the SOM canonical form will contain fewer literals than POM. However, if it has fewer ‘0’ entries then the POM form will have fewer literals than SOM.

Boolean Algebra and Logic Gates 37 Converting to Sum-of-Minterms Form  A function that is not in the Sum-of-Minterms form can be converted to that form by means of a truth table  Consider F = y + x z xyzFMinterm 0001m 0 = x y z 0011m 1 = x y z 0101m 2 = x y z m 4 = x y z 1011m 5 = x y z F = ∑(0, 1, 2, 4, 5) = m 0 + m 1 + m 2 + m 4 + m 5 = x y z + x y z + x y z + x y z + x y z

Boolean Algebra and Logic Gates 38 Converting to Product-of-Maxterms Form  A function that is not in the Product-of-Minterms form can be converted to that form by means of a truth table  Consider again: F = y + x z xyzFMinterm M 3 = (x+y+z) M 6 = (x+y+z) 1110M 7 = (x+y+z) F = ∏(3, 6, 7) = M 3 · M 6 · M 7 = (x+y+z) (x+y+z) (x+y+z)

Boolean Algebra and Logic Gates 39 Conversions Between Canonical Forms F = m 1 +m 2 +m 3 +m 5 +m 7 = ∑(1, 2, 3, 5, 7) = x y z + x y z + x y z + x y z + x y z F = M 0 · M 4 · M 6 = ∏(0, 4, 6) = (x+y+z)(x+y+z)(x+y+z) xyzFMintermMaxterm 0000M 0 = (x + y + z) 0011m 1 = x y z 0101m 2 = x y z 0111m 3 = x y z 1000M 4 = (x + y + z) 1011m 5 = x y z 1100M 6 = (x + y + z) 1111m 7 = x y z

Boolean Algebra and Logic Gates 40 Algebraic Conversion to Sum-of-Minterms  Expand all terms first to explicitly list all minterms  AND any term missing a variable v with (v + v)  Example 1: f = x + x y(2 variables) f = x (y + y) + x y f = x y + x y + x y f = m 3 + m 2 + m 0 = ∑(0, 2, 3)  Example 2: g = a + b c(3 variables) g = a (b + b)(c + c) + (a + a) b c g = a b c + a b c + a b c + a b c + a b c + a b c g = a b c + a b c + a b c + a b c + a b c g = m 1 + m 4 + m 5 + m 6 + m 7 = ∑ (1, 4, 5, 6, 7)

Boolean Algebra and Logic Gates 41 Algebraic Conversion to Product-of-Maxterms  Expand all terms first to explicitly list all maxterms  OR any term missing a variable v with v · v  Example 1: f = x + x y(2 variables) Apply 2 nd distributive law: f = (x + x) (x + y) = 1 · (x + y) = (x + y) = M 1  Example 2: g = a c + b c + a b(3 variables) g = (a c + b c + a) (a c + b c + b)(distributive) g = (c + b c + a) (a c + c + b)(x + x y = x + y) g = (c + b + a) (a + c + b)(x + x y = x + y) g = (a + b + c) (a + b + c) = M 5. M 2 = ∏ (2, 5)

Boolean Algebra and Logic Gates 42 Function Complements  The complement of a function expressed as a sum of minterms is constructed by selecting the minterms missing in the sum-of-minterms canonical form  Alternatively, the complement of a function expressed by a Sum of Minterms form is simply the Product of Maxterms with the same indices  Example: Given F(x, y, z) = ∑ (1, 3, 5, 7) F(x, y, z) = ∑ (0, 2, 4, 6) F(x, y, z) = ∏ (1, 3, 5, 7)

Boolean Algebra and Logic Gates 43 Summary of Minterms and Maxterms  There are 2 n minterms and maxterms for Boolean functions with n variables.  Minterms and maxterms are indexed from 0 to 2 n – 1  Any Boolean function can be expressed as a logical sum of minterms and as a logical product of maxterms  The complement of a function contains those minterms not included in the original function  The complement of a sum-of-minterms is a product-of- maxterms with the same indices

Boolean Algebra and Logic Gates 44  Standard Sum-of-Products (SOP) form: equations are written as an OR of AND terms  Standard Product-of-Sums (POS) form: equations are written as an AND of OR terms  Examples: SOP: POS:  These “mixed” forms are neither SOP nor POS Standard Forms B C B A C B A   C · ) C B (A · B) (A  C) (A C) B (A  B) (A C A C B A 

Boolean Algebra and Logic Gates 45 Standard Sum-of-Products (SOP)  A sum of minterms form for n variables can be written down directly from a truth table. Implementation of this form is a two-level network of gates such that: The first level consists of n-input AND gates The second level is a single OR gate  This form often can be simplified so that the corresponding circuit is simpler.

Boolean Algebra and Logic Gates 46  A Simplification Example:  Writing the minterm expression: F = A B C + A B C + A B C + ABC + ABC  Simplifying: F = A B C + A (B C + B C + B C + B C) F = A B C + A (B (C + C) + B (C + C)) F = A B C + A (B + B) F = A B C + A F = B C + A  Simplified F contains 3 literals compared to 15 Standard Sum-of-Products (SOP) )7,6,5,4,1()C,B,A(F  

Boolean Algebra and Logic Gates 47 AND/OR Two-Level Implementation  The two implementations for F are shown below It is quite apparent which is simpler!

Boolean Algebra and Logic Gates 48 SOP and POS Observations  The previous examples show that: Canonical Forms (Sum-of-minterms, Product-of-Maxterms), or other standard forms (SOP, POS) differ in complexity Boolean algebra can be used to manipulate equations into simpler forms Simpler equations lead to simpler implementations  Questions: How can we attain a “simplest” expression? Is there only one minimum cost circuit? The next part will deal with these issues

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