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\textbf{\LARGE BOOLEAN ALGEBRA}\\[20pt]
\textbf{\large PROJECT}\\[15pt]
\end{center}
\large Submitted to the University of Kerala in partial fulfillment of the requirements for the award of the Degree of Bachelor of Science.\\
\begin{center}
\Large in\\
\Large Mathematics\\[50pt]
\Large By\\
\end{center}
\textbf{\large ASWATHY J \hfill 22011100ffg004}\\
\textbf{\large DEEPA S. \hfill 220111t62387900027}\\
\textbf{\large JAYAI J.S \hfill 22011106789-0032}\\
\textbf{\large SREEJA L \hfill 22011100044}\\
\textbf{\large LAKSHMI V \hfill 220434550047}\\
\textbf{\large reshma \hfill 2201115670049}\\
\begin{center}
\textbf{\large Exam Code:22010602}\\[15pt]
Under the Guidance of Smt. SEBINA Assistant Professor, All hjsadkh College Trivandrum\\[35pt]
\textbf{\Large ALL GFGHJHJ COLLEGE}\\
\textbf{\large Thiruvananthapuram}\\
\large 2014\\[200pt]
\end{center}
\begin{center}
\textbf{\LARGE CERTIFICATE}
\end{center}
{\large This is to certify that project entitled \textquoteblleftBOOLEAN ALGEBRA\textquoteblright is based on the work carried out by Aswathy J, Deepa S , Jayai J S, Sreeja L, Lakshmi V, reshma under the guidance of Smt. Sebina , Assistant Professor in Mathematics, All gfgujhyhgh's College, Thiruvananthapuram and no part of this work has formed the basis for the award of any degree or diploma to any other university.}\\[25pt]
{\large {2014,March 17 \\ Thiruvananthapuarm}}\\[10pt]
{\large Head of the Department}\\[10pt]
{\large Teacher in charge:}\\[500pt]
\begin{center}
\textbf{\LARGE ACKNOWLEDGEMENT}
\end{center}
We are deeply indebted to my superior Smt.SEBINA,Assistant Professor, Department of Mathematics ,All hjjhgjh College, Thiruvananthapuram, for her inspiring guidance and constant help throughout the preparation of this project work.I also thank all other teachers of All Saint's College for providing us all necessary facilities to carry out this work. We also extend our thanks to all our friends who helped us in one or other way to carry out this work.\\[35pt]
\begin{flushright}
\Large ASWATHY J \\
\Large DEEPA S\\
\Large JAYA J.S\\
\Large SREEJA L\\
\Large LAKSHMI V\\
\Large reshma\\[350pt]
\end{flushright}
\begin{center}
\textbf{\LARGE CONTENTS}\\[30pt]
\end{center}
1.\Large Introduction\\
2.\Large Logic Gates\\
3.\Large Basic Gates\\
4.\Large More about Logic Gates\\
5.\Large Universal Gates\\
6.\Large Boolean Operation\\
7.\Large Basic Postulates of Boolean Algebra\\
8.\Large Principle of Duality\\
9.\Large Basic Theorems\\
10.\Large Demorgans Theorem\\
11.\Large Rules for Boolean Algebra\\
12.\Large Applications\\
13.\Large Bibliography\\[300pt]
\begin{center}
\textbf {\LARGE INTRODUCTION} \\[20pt]
\end{center}
{\large Boolean Algebra is a mathematical system of logic expressing truth functions as symbols . These symbols are manipulated to arrive at a conclusion. George Boole developed this branch of mathematics in his book \textquoteblleftAn Investigation Of The Laws Of Thoughts\textquoteblright, now known as Symbolic Logic. This provided the basic logic for operations on binary numbers. It predicted the modern developments in abstract algebra and mathematical logic. It was applied to switching circuits with immense success . Today, it forms the backbone of digital electronics.\\[10pt]In Boolean Algebra, there are only two logic variables. These two logic variables are constants within the boolean system and are represented by 0 and 1. As the logic variables have only two states, it is natural to represent them by either 0 or 1. For instance, the electric lamp has only two states,either ON or OFF. The ON state is represented by 1 and OFF state by 0. Similarly, a variable which has two states of either true or false can be described as true=1 and false=0. In the student records of a college, sex can be denoted by the logic variable X as X=1 for male and X =0 for female. It may be noted that Boolean values of 1 or 0 donot have numerical significance but denote the logic state only.}\\[320pt]
\begin{center}
\textit{\LARGE LOGIC GATES}\\[10pt]
\end{center}
{\large A gate is a digital circuit (logic circuit) with one output and one or more inputs, an output signal occuring only for certain combinations of input signals .It is conventional to represent the inputs by the first letters of the alphabets in capital letter ; A,B etc and output by X,Y etc . Thus the circuit follows a certain logical relationship between the input and the output.\\[10pt] The operation of the gate is based on Boolean Algebra . Boolean algebra is the backbone of the binary system. Boolean algebra has the advantage of being simple , speedy and accurate.\\[10pt] A Boolean function can be viewed as the description of particularv logic circuit having several inputs and one output.Such a circuit can be produced by Boolean polynomials of the type:r.s,r+s,r',etc.Logic diagrams for such polynomials can be one way of constructing a circuit having same description as boolean function under consideration . Thus,logic diagrams of these polynomials represent alternative methods for constructing the desired circuit in terms of logic gates.\\[10pt]In logic diagrams,gaqtes are represented by symbols and inputs and output by arrowed lines labelled by letters.The following are gates symbols with their names which are used in the switch theory. }\\[100pt]
\begin{center}
\textit{\LARGE BASIC LOGIC GATES}\\[20pt]
\end{center}
\Large A Gate is simply an electronic circuit which operates on one or more signals to produce an output signal.The most common logic gates are AND,OR and NOT gates.\\[20pt]
1. The OR gate[logical addition]\\[20pt]
For an OR gate there can be two or more inputs and an output. A and B are the inputs and Y is the output. Boolean expression for OR gate is
\begin{center}
A+B=Y
\end{center}
which reads A or B equals Y. The concept of addition should not be associated with the indicated symbol(+). This equation conveys only ideas and not numbers. The logic gate symbol of the OR gate is shown in the figure.\\[90pt]
\begin{center}
\includegraphics{orgate99.jpg}\\[50pt]
\end{center}
The truth table of OR gate is:
\begin{center}
\includegraphics{orgate5.jpg}\\[50pt]
\end{center}
\Large 2. The And gate [logical multiplication]\\[20pt]
\Large For an And gate there are two or more inputs and output ; A and B are the inputs and Y is the output .
Boolean expression for AND gate is A.B or AB=Y ,which reads A and B equals Y. The logic gate symbol of AND gate is shown in figure.\\[60pt]
\begin{center}
\includegraphics{orgate6.jpg}\\[40pt]
\end{center}
The truth table of AND gate is:
\begin{center}
\includegraphics{orgate7.jpg}\\
\end{center}
\Large 3. The NOT gate [logical negation]\\[20pt]
It is a circuit that has only one input and one output variable. In the simplest form, it consists of a switch A connected parallel to the output Y . An output is obtained only for LOW level of the input.An output is NOT obtained for HIGH level of the input. It is a simple input gate for which the output is always the complement of the input. Conplement of 1 is Zero and that of zero is 1. The complement of A is denoted by $ A' $or $\bar{A}$. Logic gate symbol for NOT gate is shown in figure.\\
\begin{center}
\includegraphics{orgate8.jpg}\\[50pt]
\end{center}
The truth table of NOT gate is:
\begin{center}
\includegraphics{orgate9.jpg}\\
\end{center}
\begin{center}
\textit{\LARGE MORE ABOUT LOGIC GATES}\\[30pt]
\end{center}
\large Logic gates perform basic logical functions and are the fundamental building blocks of digital integrated circuits. Most logic gates take an input of two binary values, and output a single value of a 1 or 0. Some circuits may have only a few logic gates, while others, such as microprocessors, may have millions of them.\\[10pt]
\large In the following examples, each logic gate except the NOT gate has two inputs, A and B, which can either be 1 (True) or 0 (False). The resulting output is a single value of 1 if the result is true, or 0 if the result is false.
\begin{itemize}
\item XOR - True if either A or B are True, but False if both are True
\item NAND - AND followed by NOT: False only if A and B are both True
\item NOR - OR followed by NOT: True only if A and B are both False
\item XNOR - XOR followed by NOT: True if A and B are both True or both False. \\
By combining thousands or millions of logic gates, it is possible to perform highly complex operations. The maximum number of logic gates on an integrated circuit is determined by the size of the chip divided by the size of the logic gates. Since transistors make up most of the logic gates in computer processors, smaller transistors mean more complex and faster processors.
\end{itemize}
\begin{enumerate}
\item The NOR gate\\[20pt]
The NOR gate is a combination OR gate followed by an inverter. Its output is \textquotebllefttrue\textquoteblright if both inputs are \textquoteblleftfalse\textquoteblright . Otherwise, the output is \textquoteblleftfalse\textquoteblright .The NOR (Not - OR) gate has an output that is normally at logic level \textquoteblleft1\textquoteblright and only goes \textquoteblleftlow\textquoteblright to logic level \textquoteblleft0\textquoteblright when ANY of its inputs are at logic level \textquoteblleft1\textquoteblright. The Logic NOR Gate is the reverse or \textquoteblleftComplementary\textquoteblright form of the OR gate.
\end{enumerate}
\includegraphics{boolean88.jpg}
$\Longleftrightarrow$\\
\includegraphics{prijitha1.jpg}\\[6pt]
\begin{center}
\textit{\LARGE TRUTH TABLE}\\[6pt]
\end{center}
\begin{center}
\includegraphics{truthtable.jpg}\\[40pt]
\end{center}
2. The NAND gate\\[10pt]
It is an AND gate followed by a NOT gate.NAND gate is inverted AND gate. The boolean expression of NAND gate is$ Y=\bar{AB} $.We read it as Y equals A AND B compliment.It's symbol is:\\[15pt]
\includegraphics{picture2.jpg}
$\Longleftrightarrow$\\
\includegraphics{boolean8.jpg}
\begin{center}
\textit{\LARGE TRUTH TABLE}\\[30pt]
\end{center}
\includegraphics{truthtable1.jpg}\\
3. The XOR gate\\[20pt]
The XOR ( exclusive-OR ) gate acts in the same way as the logical \textquotebllefteither/or\textquoteblright. The output is \textquotebllefttrue\textquoteblright if either, but not both, of the inputs are \textquoteblleftTrue\textquoteblright.The output is \textquoteblleftFalse\textquoteblright if both inputs are \textquoteblleftfalse\textquoteblright or if both inputs are \textquoteblleftTrue\textquoteblright . Another way of looking at this circuit is to observe that the output is 1 if the inputs are different, but 0 if the inputs are the same.\\[20pt]
\noindent The XOR gate is a special logic gate used to realise the addition of bits. It is a combination of two NOT gates and one OR gate .The block diagram of XOR gate is as shown.\\[20pt]
\includegraphics{DocP111.jpg}\\
$\Longleftrightarrow$\\[30pt]
\begin{center}
\includegraphics{boolean68.jpg}\\[40pt]
\end{center}
\begin{center}
\textit{\LARGE TRUTH TABLE}\\
\end{center}
\begin{center}
\includegraphics{project2.jpg}\\[40pt]
\end{center}
4. The XNOR gate(Exclusive NOR gate)\\[20pt]
The \lqExclusive-NOR\rq gate circuit does the opposite to the XOR gate . It will give a false output if either, but not both, of its two inputs are true. The symbol is an EXORgate with a small circle on the output . The small circle represents inversion.\\[30pt]
It is a combination of XOR gate and NOT gate.The output of this gate is the compliment of the output of the XOR gate.In this the output is high when both inputs are alike and is low when the inputs are unlike.\\
\includegraphics{boolean61.jpg}
\begin{center}
\textit{\LARGE TRUTH TABLE}\\
\end{center}
\begin{center}
\includegraphics{truthtable2.jpg}\\[40pt]
\end{center}
\begin{center}
\textit{\LARGE UNIVERSAL GATES}\\[20pt]
NOR gate and NAND gate are together called universal gates.This is because they are combinations of the three basic logic gates and all other gates can be realised from these two gates. \\[30pt]
A universal gate is a gate which can implement any Boolean function without need to use any other gate type. The NAND and NOR gates are universal gates. In practice, this is advantageous since NAND and NOR gates are economical and easier to fabricate and are the basic gates used in all IC digital logic families. In fact, an AND gate is typically implemented as a NAND gate followed by an inverter not the other way around. Likewise, an OR gate is typically implemented as a NOR gate followed by an inverter not the other way around. \\[70pt]
\end{center}
\begin{center}
\textit{\LARGE BOOLEAN OPERATION}\\[30pt]
\end{center}
\large Boolean addition and multiplication are described by the rules given below.Since there are only two variable 0 and 1,rules for boolean addition are:
\begin{center}
\begin {align*}
0+0&=0\\
0+1&=1\\
1+0&=1\\
1+1&=1\\
\end{align*}
\end{center}
\large Boolean multiplication follows the rules of binary multiplication
\begin{center}
\begin{align*}
0*0&=0\\
0*1&=0\\
1*0&=0\\
1*1&=1
\end{align*}
\end{center}
\large Boolean operators combine binary digits to yield a single digit. The basic operators AND,OR,NAND and NOR. The operator NOT complements the boolean variable\\[20pt]
\begin{center}
\textit{\LARGE BASIC POSTULATES OF BOOLEAN ALGEBRA}\\[30pt]
\end{center}
\large Boolean algebra,being a system of mathematics ,consists of fundamental laws that are used to bulid a workable framework upon which are based on the theorems of boolean algebra .These fundamental laws are known as Basic Potulates of Boolean Algebra .These postulates state basic relation in boolean algebra ,that follows:
\begin{enumerate}
\item If $A \neq 0$,then$ A=1$;\\
and If $A\neq1$,then $A=0$.
\item OR Relation (Logical addition)
\begin{center}
\begin {align*}
0+0&=0\\
0+1&=1\\
1+0&=1\\
1+1&=1\\
\end{align*}
\end{center}
\item AND Relation (Logical multiplication)
\begin{center}
\begin{align*}
0*0&=0\\
0*1&=0\\
1*0&=0\\
1*1&=1
\end{align*}
\end{center}
\item Complement Rules
\begin{center}
\begin{align*}
\bar{0}&=1\\
\bar{1}&=0\\
\end{align*}
\end{center}
\end{enumerate}
\begin{center}
\LARGE{\textbf{PRINCIPLE OF DUALITY}}\\[15pt]
\end{center}
\quad \quad \quad \quad This is a very important principle used in Boolean algebra.\, This states that Boolean relation can be derived by
\begin{enumerate}
\item Changing each OR sign (+) to an AND sign (.).\\
\item Changing each AND sign (.) to an OR sign (+).\\
\item Replacing each 0 by1 and each 1 by 0.\\
The derived relation using duality principle is called \textit{duality of original expression}.\\
For instance,we take postulates (2 )related to logical addition,which states
\begin{enumerate}
\item 0 + 0 = 0\\
\item 0 + 1 = 1\\
\item 1 + 0 = 1\\
\item 1 + 1 = 1\\
Now,working according to these guidelines,+ is changed to . and 0's are replaced by 1's, these become
\begin{enumerate}
\item 1 . 1 = 1\\
\item 1 . 0 = 0\\
\item 0 . 1 = 0\\
\item 0 . 0 = 0\\
\end{enumerate}
\end{enumerate}
\end{enumerate}
which are nothing but same as that of postulates (3) related to logical multiplication.\, So i.,\,ii.,\,iii.,\,iv. are the duals of (a),\,(b),\,(c),\,(d).\, \\[10pt]
\begin{center}
\LARGE{\textbf{BASIC THEOREMS}}\\[15pt]
\end{center}
\quad \quad \quad \quad Basic postulates of Boolean algebra are used to define basic theorems of Boolean algebra that provide all the tools necessary for manipulating Boolean expressions.\, Although simple in appearance,these theorems may be used to construct the Boolean algebra.\\
\begin{enumerate}
\item \Large{Properties of 0 and 1}\\
\begin{enumerate}
\item 0 + A = A\\[2pt]
Proof:\\
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
0&A&Y\\
\hline
0&0&0\\
0&1&1\\
\hline
\end{tabular}
\end{center}
where Y signifies the output.\, As A can have either 0 or 1 (postulate 1) both the values ORed with 0 produced the same output as that of A.\\
\item 1 + A = 1\\[2pt]
Proof:\\
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
1&A&Y\\
\hline
1&0&1\\
1&1&1\\
\hline
\end {tabular}
\end{center}
A can have values 0 or 1.\, both the values (0 and 1) ORed with 1 produced the output as 1.\\
Therefore,\,1 + A = 1 is a \textit{tautology}.\\
\item 0 . A = 0\\[2pt]
Proof:\\
\quad \, As both the possible values of A (0 and 1) are to be ANDed with 0.\\[30pt]
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
0&A&Y\\
\hline
0&0&0\\
0&1&0\\
\hline
\end{tabular}
\end{center}
Both the values of A (0 and 1) when ANDed with produced the output as 0.\\Therefore, 0 . A = 0is a \textit{fallacy}.
\item 1 . A = A\\[2pt]
Proof:\\
\quad \, Both the possible value of (0 and 1) are to be ANDed with 1.\\
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
1&A&Y\\
\hline
1&0&0\\
1&1&1\\
\hline
\end{tabular}
\end{center}
Now,both the values (0 and 1) when ANDed with 1 produce the same output as that of A.
\end{enumerate}
\quad \quad Here, properties (b) and (c) are duals of each other and properties (a) and (d) are duals of each other.
\end{enumerate}
\Large{ 2. Indempotence Law}\\[5pt]
\quad\quad\quad\quad This law states that
(a) A + A = A\\[2pt]
Proof:\\
\quad As A is to be ORed with itself only,we will prepare truth table with two possible values of A (0 and 1).
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
A&A&Y\\
\hline
0&0&0\\
1&1&1\\
\hline
\end{tabular}
\end{center}
(b) A . A = A\\[2pt]
Proof:\\
\quad Here A is ANDed with itself with possible value of A (0 and 1).\\
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
A&A&Y\\
\hline
0&0&0\\
1&1&1\\
\hline
\end{tabular}
\end{center}
\quad \quad (a) and (b) are duals of each other.\\
\Large{ 3. Involution}\\[5pt]
\quad\quad\quad\quad This law states that
$(A')'$ = A
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
A&$A'$&$A'$\\
\hline
0&1&0\\
1&0&1\\
\hline
\end{tabular}
\end{center}
\quad First column represents possible value of A,second column represents complement of A $(A')$ and the third column represent complement of$ A'$ $((A')')$ which is same as that of A.\\
\quad\quad\quad This law is also called \textit{double inversion rule}.
\Large{ 4. Complementarity Law}\\[5pt]
\quad\quad\quad\quad This law states that\\
(a) A+$(A')$=1\\[2pt]
Proof:\\
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
A&$A'$&A+$A'$\\
\hline
0&1&1\\
1&0&1\\
\hline
\end{tabular}
\end{center}
\quad Here,in the first column possible values of A have been taken,second column consist of$ A'$ values,A and$ A'$ values ORed and the output is shown in third column.\\
(b) A.$A'$=0\\
Proof:\\
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
A&$A'$&A.$A'$\\
\hline
0&1&0\\
1&0&0A\\
\hline
\end{tabular}
\end{center}
\quad A.$A'$=0,as it holds true for both the values of A.\\
\quad \quad\quad Here A.$A'$=0 is dual of A+$A'$=1. Changing (+) to (.) and 1 to 0,\\we get A.$A'$=0
\Large{ 5. Commutative Law}\\[5pt]
\quad\quad\quad\quad This law states that\\
(a) A+B=B+A\\[2pt]
Proof:\\
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
A&B&A+B&B+A\\
\hline
0&0&0&0\\
0&1&1&1\\
1&0&1&1\\
1&1&1&1\\
\hline
\end{tabular}
\end{center}
\quad Therefore,A+B and B+A are identical.\\
(b) A.B=B.A\\[2pt]
Proof:\\
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
A&B&A.B&B.A\\
\hline
0&0&0&0\\
0&1&1&1\\
1&0&1&1\\
1&1&1&1\\
\hline
\end{tabular}
\end{center}
\quad Therefore,A.B and B.A are identical.\\
\Large{6. Associative Law}\\[5pt]
\quad\quad\quad\quad The associative laws of addition and multiplication in Boolean algebra are similar to those of ordinary algebra.\,They are stated as follows\\
(a) A+(B+C)=(A+B)+C\\[20pt]
Proof:\\
\quad The laws states that ORing or ANDing of several variables give the same result regardless of the grouping of the variables.\,The law of addition applied to OR gates.\\
\begin{center}
TRUTH TABLE\\[30pt]
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
A&B&C&B+C&A+B&A+(B+C)&(A+B)+C\\
\hline
0&0&0&0&0&0&0\\
0&0&1&1&0&1&1\\
0&1&0&1&1&1&1\\
0&1&1&1&1&1&1\\
1&0&0&0&1&1&1\\
1&0&1&1&1&1&1\\
1&1&0&1&1&1&1\\
1&1&1&1&1&1&1\\
\hline
\end{tabular}
\end{center}
\quad\quad\quad Compare the columns A+(B+C) and (A+B)+C,both of these are identical.\\[20pt]
(b) A.(B.C)=(A.B).C\\[40pt]
Proof:\\[20pt]
The law of multiplication applied to AND gates.\\[70pt]
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
A&B&C&B.C&A.B&A.(B.C)&(A.B).C\\
\hline
0&0&0&0&0&0&0\\
0&0&1&0&0&0&0\\
0&1&0&0&0&0&0\\
0&1&1&1&0&0&0\\
1&0&0&0&0&0&0\\
1&0&1&0&0&0&0\\
1&1&0&0&1&0&0\\
1&1&1&1&1&1&1\\
\hline
\end{tabular}
\end{center}
\quad\quad\quad Compare the column A.(B.C) and (A.B).C,both of these are identical.\\
\, Since,rule (b) is dual of rule (a).
\Large{ 7. Absorption Law}\\[5pt]
\quad\quad\quad\quad According to this law\\
(a) A+AB=A\\[2pt]
Proof:\\
\begin{center}
TRUTH TABLE
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
A&B&AB&A+AB\\
\hline
0&0&0&0\\
0&1&0&0\\
1&0&0&1\\
1&1&1&1\\
\hline
\end{tabular}
\end{center}
\quad\quad\quad Column A and A+AB are identical.\, Also it can be proved
algebraically as,\\
\begin{align*}
L.H.S.&=A+AB\\
&=A(1+B)\\
&=A.1 [putting (1+B)=1]\\
&=A\\
&=R.H.S.
\end{align*}
(b) A(A+B)=A\\[2pt]
Proof:\\
\quad\quad\quad However, we are giving the algebraic proof of this law,
\begin{align*}
L.H.S.&=A(A+B)\\
&=A.A+AB\\
&=A+AB\\
&=A(1+B)\\
&=A.1\\
&=A\\
&=R.H.S.
\end{align*}
\quad \quad\quad\quad Since,rule (b) is dual of rule (a).\\[30pt]
\begin{center}
\textit{\LARGE DEMORGAN'S THEOREM}\\[40pt]
\end{center}
\Large One of the most powerful identities used in Boolean algebra is Demorgan's theorem . Augustes Demorgan have paved the way to Boolean algebra by discovering these two theorems . They relate the sums and products of complements.\\[20pt]
\begin{center}
\Large Demorgan's first theorem states that the complement of a sum is equal to the product of the complement.\\[20pt]
\end{center}
\begin{center}
ie,$ \overline{A+B}=\bar{A}.\bar{B}$\\[20pt]
\end{center}
\begin{center}
\includegraphics{picture45.jpg}
\end{center}
$\Longleftrightarrow$\\
\begin{center}
\includegraphics{picture6.jpg}
\end{center}
\Large Demorgan's second theorem states that the complement of a product is equal to the sum of the complement\\
\begin{center}
$\overline{A.B}=\bar{A}+\bar{B}$\\[30pt]
\end{center}
\begin{center}
\includegraphics{picture2.jpg}\\
\end{center}
$\Longleftrightarrow$\\
\begin{center}
\includegraphics{picture7.jpg}\\[50pt]
\end{center}
\begin{center}
\textit{\LARGE RULES FOR BOOLEAN ALGEBRA}\\[30pt]
\end{center}
\Large The basic rules for manipulating and simplifying Boolean expressions are listed below.\\
\begin {align*}
&1. A+0=A\\
&2. A+1=1\\
&3. A.0=0\\
&4. A.1=A\\
&5. A+\bar{A}=1\\
&6. A.A=1\\
&7. A+A=A\\
&8. A.\bar{A}=0\\
&9.\bar{\bar{A}}=A\\
&10. A+AB=A\\
&11. A+\bar{A}B=A+B\\
&12. (A+B)(A+C)=A+BC\\
\end{align*}
\begin{center}
\textit{\LARGE APPLICATIONS}\\[40pt]
\end{center}
\Large An important application of Boolean Algebra is in Switching circuit.We can apply the rules of Boole's algebra in switching circuits and can introduce switching algebra as a way to analyse and design circuits by algebraic means in terms of logic gates.The application of Boolean Algebra to circuits provided an actual physical representation for the corresponding symbolic operations.Diagrams depicting the two types of connections and the corresponding operations are as follows:\\
\begin{center}
\Large Parallel connection : X+Y\\[30pt]
\end{center}
\begin{center}
\includegraphics{picture4.jpg}
\end{center}
\begin{center}
\Large Series Connection: X.Y\\[40pt]
\end{center}
\begin{center}
\includegraphics{picture3.jpg}
\end{center}
\Large More complicated circuit with an algebraic equation is as follows:\\
\begin{center}
\includegraphics{picture5.jpg}\\[20pt]
\end{center}
\Large In circuit engineering settings today,there is little need to consider other Boolean Algebras,thus \textquoteblleftswitching algebras\textquoteblright and \textquoteblleftBoolean Algebra\textquoteblright are often used interchangeably .Efficient implementation of Boolean functions is a fundamental problem in the design of combinatorial logic circuits.Modern electronic design automation tool for VLSL circuits often rely on an efficient representation of Boolean functions known as binary discussion diagram (BDD) for logic synthesis and formal verification.\\
\Large Logic sentences that can be expressed in classical proportional calculus have an equivalent expression in Boolean algebra.Thus,Boolean Logic is sometimes used to denote proportional calculus performed in this way.The problem of determining whether the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to true is called the Boolean satisfiability problem (SAT) and is of importance to theoretical computer science.The closely related model of computation known as a Boolean circuit relates time complexity to circuit complexity.\\[300pt]
\begin{center}
\textbf{\LARGE BIBLIOGRAPHY}\\[35pt]
\end{center}
1.Computer Science +2 Text \---- Sumita Arora\\
2.Modern Digital Electronics(4th edition) \---- R.P Jain\\
3.Computer Organisation(4th edition) \---- V Carl HAMACHER\\
4.Discrete Mathematics \---- J.k. Sharma\\
5.Complimentory Physics \---- k.M.Vargheese\\
6.Internet\\
\end{document}
答案1
您的文件存在很多问题，导致无法编译。其中包括
指令
\textquoteblleft拼写错误（约 18 次）——应该是\textquotedblleft。此指令后面还应该有一个空格，即不要将其连接到下一个单词。该指令
\textquoteblright也拼写错误（同样约有 18 次） - 应该是\textquotedblright。指令
\lq后面需要有空格——不要将其直接连接到下一个单词。
除了处理这些紧迫问题之外，您还应该努力减少“视觉格式化”，更多地对文档进行“逻辑格式化”。例如，不要插入[350pt垂直空白来填充当前页面并生成分页符，只需发出指令\clearpage即可。还请注意一些印刷问题，例如留出空格后而不是标点符号之前。
为了教会自己如何用 LaTeX 写得更地道，你可能需要从LaTeX2e 的简短介绍Tobias Oetiker 等人撰写。当然，还有更多的在线资源可以帮助您成为更熟练的作家，尤其是更熟练的 LaTeX 用户。
最后说一句：你发布了一份很长的独立文档，但你实际上并没有说明什么您希望我们特别关注的问题。我假设您需要一些帮助来追踪一些语法错误。如果不是这样，请编辑您的帖子以说明您到底需要什么帮助。