# Lecture 5. Chapter 2: Sections 4-7

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1 Lecture 5 Chapter 2: Sections 4-7 Outline Boolean Functions What are Canonical Forms? Minterms and Maxterms Index Representation of Minterms and Maxterms Sum-of-Minterm (SOM) Representations Product-of-Maxterm (POM) Representations Conversions between Representations CS 2420 Husain Gholoom - lecturer Page 1

2 Boolean Algebra Boolean algebra allows us to apply provable mathematical principles to help us design logical circuits An algebraic structure defined on a set of at least two elements B={0, 1}, together with three binary operators (denoted +, and ') that satisfies the following basic identities: Properties of Boolean Algebra Closure: The structure is closed with respect to the operator +,, ' CS 2420 Husain Gholoom - lecturer Page 2

3 Boolean Functions Boolean function described by an algebraic expression consists of binary variables, 0, 1 and logic operation symbols. Boolean equations, truth tables and logic diagrams describe the same function! Truth tables are unique; expressions and logic diagrams are not. This gives flexibility in implementing functions. CS 2420 Husain Gholoom - lecturer Page 3

4 Boolean Functions F2=x'y'z+x'yz+xy' CS 2420 Husain Gholoom - lecturer Page 4

5 Boolean Functions F2=x'y'z+x'yz+xy'=x'z(y'+y)+xy'=x'z+xy' By manipulating a boolean expression, obtain a simpler expression and thus reduce the number of gates and the number of inputs (reduce the cost of a circuit) CS 2420 Husain Gholoom - lecturer Page 5

6 Expression Simplification Each term requires a gate. Literal: a single variable within a term, in complemented or uncomplemented form. F2=x'y'z+x'yz+xy' (3 terms, 8 literal) F2=x'y'z+x'yz+xy'=x'z+xy' (2 terms, 4 literal) Simplify to contain the smallest number of literals (complemented and uncomplemented variables): (x+y)(x+y') CS 2420 Husain Gholoom - lecturer Page 6

7 FUNCTION EXAMPLE Functions of Three Variables f1 = x'y'z+xy'z'+xyz = (x+y+z)(x+y'+z)(x+y'+z')(x'+y+z')(x'+y'+z) f2 = x'yz+xy'z+xyz'+xyz = (x+y+z)(x+y+z')(x+y'+z)(x'+y+z) CS 2420 Husain Gholoom - lecturer Page 7

8 Minterm Function Example Example: f1 = x'y'z + xy'z' + xyz CS 2420 Husain Gholoom - lecturer Page 8

9 Maxterm Function Example Example: Implement f1 in maxterms: f1 = (x+y+z)(x+y'+z)(x+y'+z')(x'+y+z')(x'+y'+z) f CS 2420 Husain Gholoom - lecturer Page 9

10 Overview Canonical Forms What are Canonical Forms? Minterms and Maxterms Index Representation of Minterms and Maxterms Sum-of-Minterm (SOM) Representations Product-of-Maxterm (POM) Representations Conversions between Representations Canonical Forms It is useful to specify Boolean functions in a form that: Allows comparison for equality. Has a correspondence to the truth tables Canonical Forms in common usage: Sum of Minterms (SOM) Product of Maxterms (POM) CS 2420 Husain Gholoom - lecturer Page 10

11 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., x' ), there are 2 n minterms for n variables. Example: Two variables (X and Y)produce 2 x 2 = 4 combinations: XY (both normal) XY' (X normal, Y complemented) X'Y (X complemented, Y normal) X'Y' (both complemented) Thus there are four minterms of two variables. CS 2420 Husain Gholoom - lecturer Page 11

12 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: X+Y (both normal) X+Y' (x normal, y complemented) X'+Y (x complemented, y normal) X'+Y' (both complemented) CS 2420 Husain Gholoom - lecturer Page 12

13 MAXTERMS AND MINTERMS - Minterms and Maxterms are designated with a subscript The subscript is a number, corresponding to a binary pattern The bits in the pattern represent the complemented or normal - For Minterms: o 0 means the variable is Complemented. o 1 means the variable is Not Complemented and - For Maxterms: o 0 means the variable is Not Complemented and o 1 means the variable is Complemented. CS 2420 Husain Gholoom - lecturer Page 13

14 Index Examples Four Variables Index Binary Minterm Maxterm i Pattern mi Mi a'b'c'd' a+b+c+d a'b'c'd? ? a+b+c'+d' a'bc'd a+b'+c+d' ? a+b'+c'+d' ab'cd' a'+b+c'+d abc'd? abcd a'+b'+c'+d' CS 2420 Husain Gholoom - lecturer Page 14

15 Minterm and Maxterm Relationship Review: DeMorgan's Theorem (x+y+z)'=x'y'z' and (xyz)'=x'+y'+z' Thus Mi is the complement of mi and vice-versa. Since DeMorgan's Theorem holds for n variables, the above holds for terms of n variables giving: Thus Mi is the complement of mi. CS 2420 Husain Gholoom - lecturer Page 15

16 Observations In the function tables: Each minterm has one and only one 1 present in the 2 n terms (a minimum of 1s). All other entries are 0. Each maxterm has one and only one 0 present in the 2 n terms All other entries are 1 (a maximum of 1s). CS 2420 Husain Gholoom - lecturer Page 16

17 MINTERM FUNCTION EXAMPLE Functions of Three Variables We can implement any function by "ORing" the minterms corresponding to "1" entries in the function table. These are called the minterms of the function. f1 = x'y'z+xy'z'+xyz = m1 + m4 + m7 f2 = x'yz+xy'z+xyz'+xyz = m3+ m5+ m6+ m7 CS 2420 Husain Gholoom - lecturer Page 17

18 Minterm Function Example Example: Find f1 = m1 + m4 + m7 f1 = x'y'z + xy'z' + xyz CS 2420 Husain Gholoom - lecturer Page 18

19 MAXTERM FUNCTION EXAMPLE Functions of Three Variables We can implement any function by "ANDing" the maxterms corresponding to "0" entries in the function table. These are called the maxterms of the function. f1 = (x+y+z)(x+y'+z)(x+y'+z')(x'+y+z')(x'+y'+z) = M0M2M3M5M6 f2 = (x+y+z)(x+y+z')(x+y'+z)(x'+y+z) = M0M1M2M4 CS 2420 Husain Gholoom - lecturer Page 19

20 Maxterm Function Example Example: Implement f1 in maxterms: f1 = (x+y+z)(x+y'+z)(x+y'+z')(x'+y+z')(x'+y'+z) = M0M2M3M5M6 CS 2420 Husain Gholoom - lecturer Page 20

21 Conversion Between Forms To convert between sum-of-minterms and product-ofmaxterms form (or vice-versa) we follow these steps: Change from products to sums, or vice versa. Swap terms in the list with terms not in the list. Example:Given F as before: Give the other form of the original function: CS 2420 Husain Gholoom - lecturer Page 21

22 Canonical Sum of Minterms Any Boolean function can be expressed as a Sum of Minterms (SOM). o For the function table, the minterms used are the terms corresponding to the 1's o For expressions, expand all terms first to explicitly list all minterms. Do this by ANDing any term missing a variable v with a term ( v+v') Example: Implement f=x+x'y' as a sum of minterms. o First expand terms: f=x(y+y')+x'y' o Then distribute terms: f=xy+xy'+ x'y' o Express as sum of minterms: f = m3 + m2 + m0 CS 2420 Husain Gholoom - lecturer Page 22

23 Another SOM Example Example: F = A + B'C There are three variables, A, B, and C which we take to be the standard order. Expanding the terms with missing variables: Collect terms (removing all but one of duplicate terms): Express as SOM: CS 2420 Husain Gholoom - lecturer Page 23

24 Shorthand SOM Form From the previous example, we started with: F = A + B'C We ended up with: F = m1+m4+m5+m6+m7 This can be denoted in the formal shorthand: Note that we explicitly show the standard variables in order and drop the m designators. CS 2420 Husain Gholoom - lecturer Page 24

25 Canonical Product of Maxterms Any Boolean Function can be expressed as a Product of Maxterms (POM). For the function table, the maxterms used are the terms corresponding to the 0's. For an expression, expand all terms first to explicitly list all maxterms. Do this by first applying the second distributive law, ORing terms missing variable v with a term equal to v v' and then applying the distributive law again. Example: Convert to product of maxterms: f(x,y,z)=x+x'y' Apply the distributive law: x + x' y' = (x + x') (x + y') = x + y' Add missing variable z: x+y'+z z'=(x+y'+z)(x+y'+z') Express as POM: f = M2 M3 CS 2420 Husain Gholoom - lecturer Page 25

26 Another POM Example Convert to Product of Maxterms: f(x,y,z) = xy + x'z Standard Forms 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: ABC + A'B'C + B POS: ( A + B ). ( A + B' + C' ). C These mixed forms are neither SOP nor POS ( A B + C ) ( A + C ) A B C' + A C ( A + B ) CS 2420 Husain Gholoom - lecturer Page 26

27 Standard Sum-of-Products (SOP) and Productof-Sum (POS) Implementation of SOP (POS) is a two-level network of gates such that: The first level consists of n-input AND (OR) gates, and The second level is a single OR (AND) gate CS 2420 Husain Gholoom - lecturer Page 27

28 AND/OR Two-level Implementation of SOP Expression A Boolean function maybe expressed in a non standard form: F=AB+C(D+E) It can be changed to standard form. CS 2420 Husain Gholoom - lecturer Page 28

29 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 two-level implementations Questions: How can we attain a simplest expression? Is there only one minimum cost circuit? The next part will deal with these issues. CS 2420 Husain Gholoom - lecturer Page 29

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