Boolean Algebra eBook 2023 PDF

Boolean Algebra For Students, Professionals and Beyond eBook 20 w w w. el ec t r o n i c s -t u to r i a l s.w s ...

Boolean Algebra For Students, Professionals and Beyond eBook 20 w w w. el ec t r o n i c s -t u to r i a l s.w s  B oole a n A lgebr a TABLE OF Our Terms of Use CONTENTS This Basic Electronics Tutorials eBook is focused on Boolean algebra and switching theory with the information presented within this ebook provided “as-is” for general information purposes only. 1. Introduction............................... 1 2. The Switching Theory of a Single Switch.............. 1 All the information and material published and presented herein including the text, graphics and images is the copyright or similar such rights of Aspencore. This represents 3. The Switching Theory of Series Switches............. 2 in part or in whole the supporting website: www.electronics-tutorials.ws, unless 4. The Switching Theory of Parallel Switches............. 2 otherwise expressly stated. 5. The Idempotent Law of Switches.................. 3 This free e-book is presented as general information and study reference guide for the 6. The Annulment Law of Switches................... 3 education of its readers who wish to learn Electronics. While every effort and reasonable care has been taken with respect to the accuracy of the information given herein, the 7. The Identity Law of Switches..................... 3 author makes no representations or warranties of any kind, expressed or implied, about 8. The Complement Law of Switches.................. 4 the completeness, accuracy, omission of errors, reliability, or suitability with respect to the information or related graphics contained within this e-book for any purpose. 9. The Double Complement Law.................... 4 10. The Commutative Law........................ 5 As such it is provided for personal use only and is not intended to address your particular problem or requirement. Any reliance you place on such information is therefore strictly at 11. The Distributive Law......................... 5 your own risk. We can not and do not offer any specific technical advice, troubleshooting 12. The Absorptive Law.......................... 6 assistance or solutions to your individual needs. 13. The Associative Law......................... 6 We hope you find this guide useful and enlightening. For more information about any of 14. De Morgan’s Theorem........................ 7 the topics covered herein please visit our online website at: 15. Boolean Algebra Laws........................ 8 www.electronics-tutorials.ws 16. Boolean Algebra Functions..................... 9 17. Boolean Algebra Examples...................... 9 Copyright © 2023 Aspencore www.electronics-tutorials.ws All rights reserved  B oole a n A lgebr a 1. Introduction 2. The Switching Theory of a Single Switch In 1854, George Boole performed an investigation into the “laws of thought” which were You may think that a switch is, well just a switch, that can be used to turn a light “ON” or based around a simplified version of the “group” or “set” theory. From his investigations “OFF”. But a switch can also be a complex electromechanical device used as a two-value Boolean Algebra was developed. Boolean Algebra deals mainly with the theory that logic element performing logical operations. In which its two physical states, open and closed, and set operations are either “TRUE” or “FALSE” but cannot be both at the same time. correspond to the logic values of 0 and 1 respectively as shown in Figure 1. Thus, Boolean Algebra is a form of mathematics based on symbolic logic. It has its own Figure 1. A Single Switching Circuit set of rules or laws, which we can use to analyse digital circuits and logic gates in an A Lamp Here in this single switch example of Figure 1. If attempt to reduce the number of logical devices required (and therefore cost). +V switch S1 is not-pressed and therefore open, the S1 lamp will be OFF. Likewise, if switch S1 is pressed Each variable (input or output) used in any Boolean Algebra expression can only have one of two distinct logic values, a “0” and a “1”, commonly referred to as “truth values” (hence closing it, the lamp will be ON. Under normal the term truth table). Then Boolean algebra is an algebraic system consisting of a set of steady state conditions, the switch is permanently open so the lamp is always OFF. variables (0 and 1) and implementation using operations that we call Boolean functions. We can develop this switching theory idea further by saying that when the lamp is ON or But a Boolean expression can have an infinite number of variables within it, all labelled illuminated, its switching algebra variable will be a logic-1, and when the lamp is OFF and individually to represent the individual inputs for that expression. For example, we could not illuminated, its switching algebra variable will be a logic-0. use variables A, B, C etc. to give us a logical expression of A+B = C. But again each variable Thus we can use switching algebra to describe the operation of the circuit containing the can ONLY have a value of logic-0 or a logic-1. switch in Figure 1. If we label the normally-open switch as a Boolean variable with the Switching Theory allows us to understand the Boolean Algebra is a form letter “A“, when the switch is open, that is “A” is not-pressed, we can define the value of operation and relationship between Boolean Algebra of logic algebra invented “A” as being “0”. Again, when the switch is closed, that is “A” is pressed, we can define the and two-level logic functions with regards to digital by George Boole value of “A” as being “1”. This switching algebra assumption is true for ALL normally-open logic gates whose operation can best be described with switch configurations. a Boolean expression. Table 1. Truth Table of Figure 1. Switching theory can be used to further develop the theoretical knowledge and concepts Thus, when the switch is pressed (activated) the lamp, (L) of digital circuits when viewed as a complex network of switching elements that switch Switch Lamp (A) (L) is ON, so A = 1 and L = 1. When the switch is not-pressed between zero and one to produce a defined output state or condition. (unactivated) the lamp (L) is OFF, so A = 0 and L = 0. 0 0 These logical states can be presented using electromechanical contacts in the form of 1 1 Therefore, we can correctly say that the output L = A for switches or relays as a logic circuit element. The implementation of switching functions the switching theory of the lamp as shown in the truth in digital logic circuits is nothing new, but it can give us a better understanding of how a L=A table for Figure 1. single digital logic gate works. w w w.e l e c tro nic s- tu to r ials.ws 1  B oole a n A lgebr a In Boolean Algebra terms, this expression is that of the AND function which is denoted by 3. The Switching Theory of Series Switches a single dot or full stop symbol, (.) between the variables giving us the Boolean expression of: L = A.B. We have seen in Figure 1. that the lamp circuit above can be controlled using a single switch, S1 and when S1 is closed (pressed) the lamp is “ON” representing a logic-1 Thus when switches are connected together in series their operation and switching condition. But what if we added a second switch in series with S1, how would that affect theory is the same as for the digital logic “AND” gate because if both inputs are logic-1, the switching function of the circuit and the illumination of the lamp. then the output is at logic-1, otherwise the output is at logic-0. Figure 2. Two Switches in Series 4. The Switching Theory of Parallel Switches The switching circuit of Figure 2. consists of two Lamp A B switches connected in series to a lamp across a If we now connect the two switches, S1 and S2 in parallel as shown, how would this +V voltage source, +V. To distinguish the operation S1 S2 arrangement affect the switching function of the circuit and the illumination of the lamp. of each individual switch, we shall label the first switch, S1 with the letter “A“, and label second switch, S2 with the letter “B“. Thus switches Figure 3. Two Switches in Parallel A and B represent two different Boolean variables. A The switching circuit now consists of the two When either switch is open, that is not-pressed, we can define the value of A as being at switches in parallel with the voltage source and S1 Lamp logic-0 and B as also being at logic-0. Likewise, when either switch is closed or pressed, the lamp. we can define the value of A as being logic-1 or B as being at logic-1. The same as before. +V B As before, with two switches, A and B, there As there are two switches, A and B, we can see that there are four possible combinations are four possible combinations of the Boolean S2 of the Boolean variables A and B to illuminate the lamp. For example, A is open and B is variables required to illuminate the lamp. That closed, or A is closed and B is open, or both A and B are open or closed at the same time. is A is open and B is closed, or A is closed and B is open, both A and B are open, or both Then we can define these operations in the following truth table of Table 2. closed at the same time. As shown in the following truth table of Table 3. Table 2. Switching Truth Table of Figure 2. Table 3. Switching Truth Table of Figure 3. Table 2. shows that the lamp will only be “ON” and Table 3. shows that the lamp will only be “ON” and Switch Switch Lamp illuminated when BOTH switch, A AND switch, B are Switch Switch Lamp (A) (B) (L) (A) (B) (L) illuminated when EITHER switch, A OR switch, B are pressed and closed as pressing only one switch on its pressed and closed. 0 0 0 own will not illuminate the lamp. 0 0 0 0 1 0 0 1 1 This therefore shows that when the two switches A and This proves that when two Boolean variables A and 1 0 0 1 0 1 B are connected in parallel, the will lamp illuminate B are connected in series, the only condition that will when any one of the switches, or both are closed. This 1 1 1 illuminate the lamp is when both switches are closed 1 1 1 gives the Boolean expression of: L = A OR B. L = A AND B giving the Boolean expression of: L = A AND B. L = A OR B w w w.e l e c tro nic s- tu to r ials.ws 2  B oole a n A lgebr a In Boolean Algebra terms, this expression is that of the OR function which is denoted by an addition or plus sign, (+) between the variables giving us the expression of: L = A+B. 6. The Annulment Law of Switches Thus when switches are connected together in parallel their switching theory is the same Boolean algebra expressions can also consist of operations using 0 and 1 in the form as for the digital logic “OR” gate because if both inputs are logic-0, then the output is 0, of postulates. While not Boolean laws in their own right, they can still be used in the otherwise the output is at logic-1. simplification of Boolean Expressions. The Annulment Law states that anything AND’ed with a 0 will always equal 0 as shown in 5. The Idempotent Law of Switches Figure 6. Also, anything OR’ed with a 1 will always be equal to a 1 as shown in Figure 7. Figure 6. Annulment Law of the AND Function Thus far we have seen how to connect two switches together either in series or parallel to illuminate a lamp. But what if the two switches representing a Boolean AND function A 0 or the Boolean OR function (operations of multiplication and sum) are of the same single A.0 = 0 Boolean variable, A. Figure 7. Annulment Law of the OR Function In Boolean Algebra there are various laws and theorems which can be used to define A the mathematics of various logic circuits. One such theorem in which combinations of a single variable are connected with itself is known by the name of the idempotent law. A+1 = 1 1 Idempotent Laws used in switching theory states that AND-ing or OR-ing a variable with itself will produce the original variable. For example, variable “A” AND’ed with “A” can be reduced to a single element, A. Likewise the Boolean variable “A” OR’ed with itself will 7. The Identity Law of Switches also result in a single element, A. Thus allowing us to simplify our switching circuits. The Identity Law states that anything AND’ed with a 1 will always equal itself as shown in Figure 4. Idempotent Law of the AND Function Figure 8. While anything OR’ed with a 0 will always be equal to itself as shown in Figure 9. Lamp A A Figure 8. Identity Law of the AND Function +V A.A = A A 1 A.1 = A Figure 5. Idempotent Law of the OR Function Figure 9. Identity Law of the OR Function A Lamp A +V A+A = A A+ 0 = A A 0 w w w.e l e c tro nic s- tu to r ials.ws 3  B oole a n A lgebr a As well as identity AND and OR operations consisting of a single Boolean variable with Figure 10. Complement Law of the AND Function the property of an element 0 or a 1, these two binary postulates (postulates are elements A taken as facts, so no proof is required) can also be used together and simplified as shown. A A.A = 0 AND Function Identity Elements Figure 11. Complement Law of the OR Function 0.0 = 0 Thus a 0 AND’ed with itself is always equal to 0 A 1.1 = 1 Thus a 1 AND’ed with itself is always equal to 1 A+A = 1 A 1.0 = 0 Thus a 1 AND’ed with an element 0 is equal to 0 OR Function Identity Elements The Boolean Law of complementation, or inversion, A.A = 0 and A +A = 1 is an important 0 + 0 = 0 Thus a 0 OR’ed with itself is always equal to 0 law to both understand an use in the simplification of Boolean expressions. 1 + 1 = 1 Thus a 1 OR’ed with itself is always equal to 1 9. The Double Complement Law 1 + 0 = 1 Thus a 1 OR’ed with an element 0 is equal to 1 As well as producing one complement of a Boolean variable, it is also possible during Note that a logic element “0” can be viewed as an additive identity. While a logic element the simplification of long Boolean expressions to end up with a double complement, or “1” can be viewed as a multiplicative identity. double negation (or more) situation. 8. The Complement Law of Switches The Double Complement Law states that if you complement or invert a variable two times, you will end up with the variables original Boolean value. That is the complement Complements (or negation, or inversion) of a Boolean variable also exists to produce the of the complemented of a variable is always equal to the variable, as two negatives make opposite of its value. The complement of a variable is represented by the NOT operation a positive since it is analogous to multiplying by -1 in ordinary or real-number algebra. which is commonly denoted with a bar (−) symbol above it. Thus if we take the variable A and complement (or invert) it once, we get not-A presented So for example, the complement of the Boolean variable A would be A. Likewise, the as A. If we then take A and complement it again, we get NOT(not-A) presented as A which complement of B would therefore be B, and so on. So, if the variable A has a value of 0, represents a double complement, which is the original variable as shown. then the complement of A would be 1, and the complement of 1 would be 0. Note that A and A’ can both be used interchangeably to represent the complement of a variable. If A = 0, then A = 1 therefore A = 0 = A. Likewise, if A = 1, then A = 0 therefore A = 1 = A The Complement Law states that any variable AND‘ed with its complement will always be That is: A is always equal to A, or could also be: A = A. Note that the complement of any equal to 0 as shown in Figure 10. If a variable is OR’ed with its complement, it will always given Boolean variable can be taken repeatedly, but will always result in the original value be equal to 1 as shown in Figure 11. every even number of times it is complemented (inverted). w w w.e l e c tro nic s- tu to r ials.ws 4  B oole a n A lgebr a Similarly, in Boolean Algebra, the AND (logical multiplication) operator will distribute over 10. The Commutative Law an OR (logical addition) operator. So, for example: A.(B+C) is the same as saying A.B + A.C as shown in Figure 13. The Commutative Law states that the order of application of two separate variables is not important as it will not affect the result of an AND or OR operation. Figure 13. Distributive Law of the AND Operator Since A and B in series is identical in all respects as B and A in series. Thus, the order in B A B which two (or more) variables are AND’ed together makes no difference as their Boolean A multiplication is commutative as shown in Figure 11. A(B + C) = AB +AC C A C Figure 11. Commutative Law of the AND Function A B B A A.B = B. A Since both switching networks produce the same result, they are therefore electrically equivalent. Thus we have distributed the AND operator with the two OR operators and Likewise, if A and B are in parallel, it is the same as B and A are in parallel as the order in then OR-ing at the end. which two (or more) variables are OR’ed makes no difference since their Boolean addition Generally, the multiplication AND operator would take priority mathematically over the is commutative as shown in Figure 12. addition OR operator. However, in Boolean Algebra the OR operator can still distribute Figure 12. Commutative Law of the OR Function evenly over an AND operator. For example: A+(B.C) would become (A+B).(A+C) as shown in Figure 14. A B Figure 14. Distributive Law of the OR Operator A+B = B +A B A A A A A+(B.C) = (A+B).(A+C) B C B C 11. The Distributive Law Here in Figure 14. we have distributed the OR operator with the AND operator for B and C While some of the previous laws of Boolean algebra may have seemed rather self-evident, and then AND-ing at the end producing two identical switching networks. or obvious. The distributive laws of AND and OR are not quite so clear. Then factorisation of any Boolean expression can be achieved by the application of the The Distributive Law permits the multiplying or factoring out of a Boolean expression distribution law allowing an expression to be expanded out by multiplying one term by in much the same way as for ordinary real-number algebra. For example, in standard the other term. arithmetic the expression 3(2 + 4) is equal to saying (3x2)+(3x4) as the final result of 18 is the same for both equations since the multiplication operator distributes over the addition operator. w w w.e l e c tro nic s- tu to r ials.ws 5  B oole a n A lgebr a 12. The Absorptive Law 13. The Associative Law The Absorptive Law does not have an exact equivalent in conventional real-number The Associative Law is another laws which governs the manipulation and simplification algebra but enables the reduction of a complicated Boolean expression to a much shorter of Boolean expressions. It allows the removal of brackets from an expression and the or simpler one by absorbing one or more like terms. In other words, the absorption law regrouping of the variables without changing the meaning of the expression. allows for the manipulation and simplification of Boolean expressions by absorbing some terms into other terms, thereby making them easier to understand and analyse. There are two versions of the Boolean associative law: one for logical OR operations and one for logical AND operations. Then we can associate terms by using the AND or OR While the law of absorption (sometimes known as the Redundancy Law) is the result of function. the application of several previous laws, nevertheless it can still be used to good effect. For example, If we have a two term expression A.B, and we want to add a third term to For example assume the following Boolean expressions of: A+(A.B) and A(A+B). produce A.B.C. Since A and the B are already grouped or associated, all we have to do is add C to the result. A + (A.B) = A.1 + A.B = A(1+B) = A.1 = A 12.1 Associative Law for the Logical AND (.) Function A(A + B) = A.A + A.B = A + A.B = A(1 + B) = A.1 = A (A.B)C = A(B.C) = A.B.C Other laws of absorption in Boolean algebra include: (A.B.C)D = (A.B).(C.D) = A.B.C.D A(A + B) = A.A + A.B = 0 + A.B = A.B Note that the associate law of multiplication is, by default associative, which means that A.B + B = A + B the product of three or more terms AND’ed together is the same in whatever order they are grouped together. A + (A.B) = A + A.B + A.B = A + B(A + A) = A + B.1 = A + B 12.2 Associative Law for the Logical OR (+) Function Then we can see that in all cases, the output is a simplification of the initial expression reducing them to their simplest form. The advantage of the absorption law is that, it helps A + (B + C) = (A + B) + C = A + B + C identify and remove redundant terms. So, if a smaller Boolean term appears within a much larger expression, then the larger term effectively becomes redundant as shown. (A + B + C) + D = (A + B) + (C + D) = A + B + C + D A + A.B + A.B.C + A.B.C + A.B.C = A For the OR associate law of addition, the order does not matter providing that the resulting circuit only contains the same type of OR gates. Indicating that the OR gate Here, the letter “A” is repeated in each term reducing down to just A. inputs can be interchanged and if required, the OR function can be extended to include three or more input variables. In all cases, if necessary the law of absorption can be proven by means of truth tables. w w w.e l e c tro nic s- tu to r ials.ws 6  B oole a n A lgebr a 13.2 De Morgan’s Second Law of AND to OR 14. De Morgan’s Theorem The second law states that the complement of the logical AND of two variables is equal to De Morgan’s Theorem is a fundamental law in Boolean algebra, which provides a the logical OR of their complements. That is, two separate terms NAND‘ed together is the relationship between the complement of logical operations allowing us to simplify an same as the two terms inverted (Complement) and OR‘ed. For example: A.B = A+B, and expression. we can prove this operation in Table 5. De Morgan’s theorem uses two sets of rules or laws to solve various Boolean algebra Table 5. Proving De Morgan’s Second Law expressions. One for the complement (negation) of the logical OR operation changing OR’s to AND’s, and one for the complement of the logical AND operation AND’s to OR’s. Inputs Truth table Output For Each Term We have seen previously that Boolean algebra uses a set of laws and rules to define the B A A.B A.B A B A+B operation of a digital logic circuit with 0’s and 1’s being used to create truth tables and mathematical expressions to define the digital operation of a logic AND, OR and NOT (or 0 0 0 1 1 1 1 inversion) operations. 0 1 0 1 0 1 1 13.1 De Morgan’s First Law of OR to AND 1 0 0 1 1 0 1 The first law states that the complement of the logical OR of two variables is equal to the 1 1 1 0 0 0 0 logical AND of their complements. That is two separate terms NOR‘ed together is the same as the two terms inverted (Complement) and AND‘ed. For example: A+B = A.B as shown in That is, the negation of the AND of two variables is equivalent to the OR of their individual Table 4. negations. Table 4. Proving De Morgan’s First Law Although we have used DeMorgan’s theorems with only two input variables A and B, they are equally valid for use with three, four or more input variable expressions, for example: Inputs Truth table Output For Each Term For a 3-variable input B A A+B A+B A B A.B 0 0 0 1 1 1 1 A.B.C = A+B+C and also A+B+C = A.B.C 0 1 1 0 0 1 0 For a 4-variable input 1 0 1 0 1 0 0 A.B.C.D = A+B+C+D and also A+B+C+D = A.B.C.D 1 1 1 0 0 0 0 and so on. In other words, the negation of the OR of two variables is equivalent to the AND of their The De Morgan’s two theorems allows us to simplify complex Boolean expressions, individual negations. rewriting them in alternative forms, or convert between different representations. w w w.e l e c tro nic s- tu to r ials.ws 7  B oole a n A lgebr a 15. Boolean Algebra Laws A in parallel with B A A A+ B = B +A = B in parallel with A B Q Commutative B The previous laws, rules and theorems for Boolean Algebra are summarised in Table 5. Table 5. Switching Equivalents of Boolean Expressions A in series with B A B A Q A.B = B. A = B in series with A B Commutative Boolean Equivalent Equivalent Boolean Algebra Description Expression Switching Circuit Logic gate Law or Rule A invert and replace OR A Q de Morgan’s A+ B = A.B with AND B 1 B Theorem A in parallel with A+1 = 1 A Q Annulment closed = CLOSED A A invert and replace AND A Q de Morgan’s A.B = A+ B with OR B A B Theorem A in parallel with Q A+ 0 = A open = A A 0 Identity The basic Boolean algebra laws which relate the Commutative Law to a change in position A in series with A 1 for addition and multiplication. The Associative Law for the removal of brackets for A.1 = A Q Identity closed = A A addition and multiplication. The Distributive Law allowing the factoring of an expression, are all the same as in ordinary algebra. A in series with A A A.0 = 0 Q Annulment Each of the Boolean Laws outlined in Table 5 are presented with just one single or two open = OPEN 0 input variables, but the number of variables defined by a single law is not limited to just A two, since there can be an infinite number of variables as inputs too any expression. A+A = A A in parallel A Idempotent with A = A Q When Boolean expressions are implemented using logic gates, each term requires a gate, A and each variable within that term represents an input to that gate. For 2-input logic A in series A A A gates, the basic logical operations where A and B are logic (or Boolean) input binary A.A = A Q Idempotent with A = A variables, and whose values can only be either logic-0 or logic-1. Which produces four possible input combinations of: 00, 01, 10, and 11 as shown in table 6. NOT NOT A = A A NOT(not-A) = A A A Double Negation (double negative) Table 6. Truth table For Each Logical Expressions A B A AND NAND OR NOR NOT Ex-OR Ex-NOR A+A = 1 A in parallel with A Complement NOT-A = CLOSED Q 0 0 0 1 0 1 1 0 1 A 0 1 0 1 1 0 0 1 0 A.A = 0 A in series with A A A Complement 1 0 0 1 1 0 1 1 0 NOT-A = OPEN Q 1 1 1 0 1 0 0 0 1 w w w.e l e c tro nic s- tu to r ials.ws 8  B oole a n A lgebr a 16. Boolean Algebra Functions 17. Boolean Algebra Examples Using the information we have learnt in this Boolean Algebra eBook, simple 2-input AND, Using the previous rules and theorems, we will practice here with a few worked examples OR and NOT Gates can be represented by 16 possible functions as shown in Table 7. as a guide. Table 7. Boolean Algebra Functions table Example 17.1 Simplify the following Boolean expression: (A +B)(A +C) Function Description Expression (A+ B).(A+C) 1. NULL 0 (A.A) + A.C + A.B + B.C Distributive law (A.A) = A Idempotent Law 2. IDENTITY 1 (A + A.C) + A.B + B.C Reduction 3. Input A A (A + A.C) = A Absorption Law 4. Input B B (A + A.B) + B.C Distributive Law 5. not-A A (A + A.B) = A Absorption Law 6. not-B B A + B.C 7. A AND B (AND Gate) A.B Thus the Boolean expression of (A + B)(A + C) can be reduced to just A + B.C using the various Boolean Algebra laws. 8. A AND not-B A.B Example 17.2 Find the Boolean expression representing the switching circuit of Figure 15. 9. not-A AND B A.B Show the resulting switching circuit and its digital logic gate equivalent. 10. NOT AND (NAND Gate) A.B Figure 15. Complex Switching Circuit 11. A OR B (OR Gate) A+B B 12. A OR not-B A+B A 13. not-A OR B A+B A B A C E +V Y 14. NOT OR (NOR Gate) A+B C D 15. Exclusive-OR A.B+A.B 16 Exclusive-NOR A.B+A.B The initial Boolean expression is determined as being: Y = A.B(A +C)(B +AC + D)E w w w.e l e c tro nic s- tu to r ials.ws 9  B oole a n A lgebr a A.B(A + C)(B + AC+D )E End of this Boolean Algebra and Switching Theory eBook A.B(B +A.C+ D )E.A+ A.B(B+ A.C+ D )E.C Distribution A.A = A Idempotent Law Last revision: April 2023 Copyright © 2023 Aspencore A.B.E(B + A.C+D ) +A.B.E(B+ A.C+ D )C https://www.electronics-tutorials.ws A + A.B = A Absorption Law Free for non-commercial educational use and not for resale A.B.E(B +A.C+ D ) A.B.E.B + A.B.E.A.C+A.B.E.D Distribution With the completion of this Boolean Algebra and Switching Theory eBook you should have A.A = A Idempotent Law gained a good and basic understanding and knowledge of the various rules and laws of A.B.E +A.B.E.A.C+ A.B.E.D Boolean to help both reduce and simplify a complex Boolean expression. The information A.A = A Idempotent Law provided here should give you a firm foundation for continuing your study of electronics A.B.E +A.B.E.C +A.B.E.D and electrical engineering as well as the study of digital logic switching circuits. A + A.B = A Absorption Law For more information about any of the topics covered here please visit our website at: A.B.E +A.B.E.D A + A.B = A Absorption Law www.electronics-tutorials.ws Y = A.B.E Main Headquarters Central Europe/EMEA Thus, the switching circuit of Figure 15. can be reduced to A.B.E using the various laws 245 Main Street Frankfurter Strasse 211 and is given along with its logic gate equivalent of a 3-input AND gate in Figure 16. Cambridge, MA 02142 63263 Neu-Isenburg, Germany Figure 16. Reduced Switching Circuit www.aspencore.com [email protected] A B E A +V Y = ABE B Y E Switching Circuit Equivalent 3-input AND Gate w w w.e l e c tro nic s- tu to r ials.ws 10