PRESENTATION ON BOOLEAN

ALGEBRA

by : SUKHDEEP SINGH

INTRODUCTIONBoolean Algebra has a set of two operators +

(similar to logical OR) and . (similar to logical AND). Any valid Boolean Algebra should satisfy

following postulates :- 1. Closure. The boolean system is closed with

respect to a binary operator if for every pair ofboolean value, it produce a boolean result. Eg. . is

closed in the boolean system because it accepts only boolean operands and produce only boolean

result. 2.Identity. A boolean value I is said to be identity

element with respect to: + : if x+I =I+x = x and . : if x.I = I.x

= x 3.Commutativity. If A and B are boolean

expression then : A.B = B.A (commutative w.r.t .) and A+B = B+A (commutative w.r.t to +) For all possible boolean value of A and B. 4.Distribution. . is distribution over + if x.

(y+z) = (x.y)+(x.z) and

+ is distributive over . If x+(y.z) = (x+y).(x+z)

5. Associativity. A binary operator is said to be associative if (A.B).C = A.(B.C) w.r.t to . and (A+B)+C = A+(B+C) w.r.t + for all boolean value A,B and C. Postulates of boolean algebra can be verified by using this table.

Basic Theorem of Boolean Algebra (a)x+x= x (f) (x+y)’ = x’.y’(b)x.x = x (g) (x.y)’ = x’+y’(c)x+1 = 1 (h) x+xy = x(d)x.0 = 0 (i) x(x+y) = x(e)(x’)’ = x

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CANONICAL FORM For any given boolean function there exist a unique canonical form.e.g. if a function F =AB+CThen equivalent canonical form can be obtained as follows : F = AB+C = AB(C+C’) + C(A+A’) = ABC + ABC’ + AC + A’C =ABC + ABC’ + AC(B+B’) + A’C(B+B’) =ABC + ABC’ + ABC +AB’C + A’BC + A’B’C = ABC + ABC’ + AB’C + A’BC + A’B’CBuilding truth table from Canonical form.1. Convert each term into a binary value by substituting a “1” for primed variable and a “0” for unprimed value.2.Place a “1” in the corresponding position in the truth table.3. Finally put “0” in all the entries that you did not fill “1” in the first step above.

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e.g. To convert F = AB + C into Truth table. Canonical form for that is : F = ABC + ABC’ + AB’C + A’BC + A’B’C1. Convert minterm to binary equivalents : F = ABC + A’BC + AB’C +A’B’C + ABC’ = 111 + 011 + 101 + 001 + 1102. Substitute a 1 in the truth table for each entry above and 0 in

the remaining .

STANDARD FORMS : There are two type of standard forms:Sum of product form (Product term are joined by OR

operator F = xy +z + x’yz )Product of sum form (Sum term are joined by AND operator

F = (x+y).(y+z’) )

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KARNAUGH MAPS (K - MAP) : Used to reduce a given boolean function to its optimal form.Examples of K map:- solving tow variable function with K – map

There are two possible combination represented by I and II in the map. Common variable value in combination I is B which correspond to value 0 in B and in combination II is A which correspond to value 0 in A. We can get result by ORing these term, so the result is –Z = A + B

Solving Three variable function with K – Map Z = f(A,B,C) = A ̅B̅C ̅ + AB + ABC + AC = A ̅B ̅C̅ + ABC + ABC +ABC + ABC + ABC

Sol. is F = B + AC + AC

Production of Sums Simplificatione.g. F(A ,B ,C ,D) = Σ(0,1,2,5,8,9,10)Solution :Creating a four variable K-map for above function , we have

In this K–map combination of 0s are made to get solution for complement of F. so, F’ = AB + CD + BD’Now F = (F’)’ = (AB + CD + BD’)’ = (A’ + B’) . (C’ + D’) . (B’ + D)Which is required product of sum simplification.

Don’t Care Conditions Till now we have discussed function in which the combination of variable are either 0 or 1 and accordingly map created. The combination usually taken from the truth table for which the value is evaluated to 1. value for all other combination is assumed to be 0. however this not always true. There may be some certain application where for certain combination, there are no output value e.g. in some digital circuit, if we have to take six input combination, we have to take 3 input variables leading to total of 2³ i.e. 8 combinations. So out of these 8 combinations 2 will be unused. So we don’t care about these value. These don’t care condition can be used in map to have simpler structure. Don’t Care combination on a K-Map are marked with ‘X’ to distinguish them from value 0 or 1. Following point should be taken while making combination with don’t care squares-1. If a combination with equal no. of squares is possible with higher no. of 1’s then that combination is given preference over the combination that contain more no. ‘X’ squares.2. It is not necessary that all squares containing ‘X’ take part in making combinations.3. Each combination containing X must have atleast one 1.

Example : Simplify F (w, x, y, z) = Σ(1, 3, 7, 11, 15) D(w, x, y, z) = Σ(0, 2, 5)Solution :Creating a four variable K-map –

Finding common value in combination, the solution will beF = wz + yz

IMPLEMENTING BOOLEAN EXPRESSION AS DIGITAL CIRCUITSFor any Boolean function you can design an electronic and vice versa. Since Boolean function only require the AND , OR, and NOT boolean operators, which will be used to represent AND (.), binary OR (+) and compliment (‘) respectively in Boolean algebraic functions.

Using these gate we can design logical diagram for any of the Boolean function.

E.g. Design logical diagram for function F = x’y + xy’ + z’

As we can conclude that all the circuits can be design with the help of these gates, we would still like to construct mare gates to gain simplicity in designing the circuit.

Some other logic gates which are very commonly used in designing logic circuit are given as below :

NAND and NOR gates as Universal GatesOne interesting fact is that you only need a single gate type to implement any electronic circuit. There are two gates which serve this purpose. There are NAND gate and NOR gate that is why these gate also known as universal gates.Converting A Function Into Logic Diagram With Only NAND Gates : Following step s have performed for doing this:1. Get the simplified form of function in sum of product form.2. Draw a NAND gate for each product term of the function that has at least two variable . These will become first level gate.3.Draw a single NAND gate in second level that have input as output from first level gate.4. Term with single variable can be complemented and applied as input to second level NAND gate.

Now using same rule we will convert AND, OR And NOT into NAND gate circuits. Remember function for these gates are already in simplified form.

NOT, AND, & OR gates equivalent to NAND gate :

E.g. F = (AB+CD) . BCLet us draw logical diagram for this

Substituting AND & OR gates equivalent to NAND gate

Implementing Logic Circuit Using NOR Gate OnlyFor two level circuit, following step are performed :1. Simplify function in product of sum form.2. Apply same step as two level NAND implementation

except term for firs level NOR gates are sumterms.3. Similarly, a single variable term will need one input

NOR gate. NOT, OR & AND gates equivalent to NOR gate:

Thank you!!!….Submitted to : MRS. SWETA

Submitted by : SUKHDEEP SINGH

Roll No. : 3607/12, Class : B.Sc. II (C.S.)