Boolean Function & Logic Circuits & Truth Table

We have learned from the Basic Boolean Expressions of NOT, OR, NOR, AND, NAND, XOR and XNOR, that they have corresponding Logic Gates for their implemention.   We can say that there are 7 Basic Gates (AND-OR-NOT-NAND-NOR-XOR-XNOR) and 3 Fundamental Gates (AND-OR-NOT).

In other words, it means that given any Boolean expression we can translate or derive the required Logic Gates for its implementation.

Below shows the Basic Boolean Functions and the Corresponding implementation of Logic Gates and Truth Tables.

The TRUTH TABLEs above describe the complete functions and the overall behavior of Logic Gates,  It can actually describe the complete functions and behavior of any Digital Logic Circuit and it can be used to compare and verify if any 2 Boolean Functions or Logic circuits are equal or similar. Hence Whenever you have a combinational logic circuit and you want to know how it works, the best way to analyze it is to use a truth table.

NOTE: we have to specify all possible input combinations in a TRUTH TABLE which can be computed as 2n.  Where n is the number of Inputs and 2n is the total number of combinations possible. For 2 inputs there are 4 possible input combinations, for 3 inputs there will be 8 combinations and for 4 inputs there are 16 combinations.

Extension to Multiple Inputs

A gate can be extended to have multiple inputs if its binary operation is commutative and associative. AND and OR gates are both commutative and associative.

For the AND function, 

            AB = BA                                  -commutative

and

            (AB)C = A(BC) = ABC.         -associative

For the OR function, 

            A + B = B + A                         -commutative

and

            (A + B) + C = A + (B + C).   -associative

These indicate that the gate inputs can be interchanged and these functions can be extended to three or more variables very simply as shown in Figures below.

To make sense on how any Boolean expression can be translated or implemented to its equivalent logic circuit we  need to know what each gate means. Remember the 3 MAIN gates (AND - OR - NOT) and the other Logic Gates it can form (NAND-NOR-XOR-XNOR).  We can say that there are 7 Basic Gates (AND-OR-NOT-NAND-NOR-XOR-XNOR) and 3 Fundamental Gates (AND-OR-NOT).

Example 1: To understand this circuit below, we must know exactly what the symbols mean. 

The circuit above can be translated as follows: 

The uppermost gate is AND and it takes as input A along with B. (AB), 

there is a ~C (Invert C) involved below it, 

and the bottom gate is an OR which takes the values of Balong with C to become (B+C). 

The results from all three of these gates becomes the input to the final AND gate giving the final output r.

The final interpretation is shown below. This gives: R = (AB)(~C)(B+C) = (AB)(C')(B+C)

Once we have the Boolean expression for a circuit output, we can obtain the output logic level for any set of input levels.

How do we evaluate or determine the output R if we apply some values to our inputs A,B,C ? 

Examining the circuit above, we can ask a number of different types of questions. For example, we can be asked to determine the value of R, given information about A, B, and C. So, If A = 1, B = 0, and C = 0, what is R?

To solve this problem we would trace the circuit writing down the values as they enter and leave each gate. Trace the circuit below and determine the values going into each gate along with the values coming out. 

As you can see if A = 1, B = 0, and C = 0 then R = 0.  

The evaluation of the above circuit is only true for that particular Input Values and we have to repeat the tracing process for all possible input combinations. To trace the circuit for all its input combinations will be tedious, but we can also get the same result and complete evaluation easily by creating its TRUTH TABLE based on the derived Boolean function of the circuit as shown below.  We had already derived the Boolean Function of the above circuit as  

R= (AB)(C')(B+C).

As mentioned before,  the TRUTH TABLE above completely describe the complete and overall function and behavior of the above Logic Circuit.   

Example 2: It does not matter how complex the circuit looks, the process of tracing the circuit and determining the logic sentence is the same. Look at this more complex circuit: 

As before, translating this circuit into the logic statement it represents means carefully using the letters A, B along with C to represent the variables going into each gate. The leftmost gate is an OR gate and all three variables are input (A+B+C), the uppermost gate is also an OR gate and it takes as input B along with (A+B+C), giving (B+(A+B+C). In the middle of the circuit, the NOT gate flips the value of (A+B+C) giving ~(A+B+C). The bottom-most AND gate take ~C as input along with ~(A+B+C) giving (~C)(~(A+B+C)).

Finally, the right-most AND gate takes as input (B+(A+B+C)), along with ~(A+B+C) and (~C)(~(A+B+C)) to give R. 

R = (B+(A+B+C)) (~(A+B+C)) (~C(~(A+B+C)))  

R = (B+(A+B+C)) (A+B+C)' (C'(A+B+C)')  

Below is its TRUTH TABLE.

Example 2B:

Give a complete Analysis of the logic circuit using truth tables.

EXERCISE -05-A:  Write the Logic statement that represents the following Logic circuits. 

d)

e)

IMPLEMENTING CIRCUITS FROM BOOLEAN EXPRESSIONS

When the operation of a circuit is defined by a Boolean expression, we can draw a logic-circuit diagram directly from that expression. For example, if we needed a circuit that was defined by  X=A.B.C we would immediately know that all that was needed was a three-input AND gate. If we needed a circuit that was defined by X = A+B' , we would use a two-input OR gate with an INVERTER on one of the inputs.The same reasoning used for these simple cases can be extended to more complex circuits.

Suppose that we wanted to construct a circuit whose output is y = AC+BC'+A'BC.  This Boolean expression contains three terms which are ORed together.  This tells us that a three-input OR

gate is required with inputs that are equal to AC+BC'+A'BC.

Constructing a logic circuit from a Boolean expression.

Example 3: Given the Booelan Function, R = (A+B+C) + (C٠B٠(~C)) + C' , derive and draw the Logic Circuit for implementation. 

Solution

Below is the TRUTH TABLE

EXERCISE 2: Given the following Booelan Functions,  derive and draw their Logic Circuits for implementation.

a)  F = xy + x'y' + y'z

b)  F = (x' + y')' + (x + y)' + (y + z')' 

c)  F = [ (xy)' (x'y')' (y'z)']'

Example-05-B: Prove that the Boolean Expression of de Morgan´s Theorem  using (3-Variable form) 

                             A)  (x + y + z)'  = x' y' z'     and  B)  (xyz)'  = x' + y' + z'  are equal.  

Using their TRUTH TABLE, we coud prove the equality of the 2 Boolean Functions or expressions of  de Morgan´s Theorem  1 and 2.      

A)

B)

                C)  Prove by TRUTH TABLE that the following Boolean  functions are equal.

                                  Fa = X' Y' Z + X' Y Z + X Y'                        Fb = X' Z + X Y'

From the examples above we now see how the TRUTH TABLE could prove the EQUALITY of 2 or more Boolean Functions.

Homework-08: (1 week)

I - Using  TRUTH TABLE, prove the equality of the following Boolean Functions.

A)  (X+Y) (X'+Z) = XZ+X'Y 

B)  (X+Y)(Y+Z)(X'+Z) = (X+Y)(X'+Z)

C)  XY+YZ+X'Z  = XY+X'Z

D) (A+C)(AB+C') = AB+AC'

E)  W'XY+WZ = (W'+Z)(W+XY)

II-

III -

A) 

B)

C) Given the Logic Circuit below, Complete the Outputs Waveforms.

      HINT,  You need to derive the TRUTH Table for the Boolean Functions

       Y and Z Outputs. Show your Truth Table.

D)  Simplify the Function below.