CMPEN 270 Laboratory 3 

CMPEN    270    Laboratory    3 

In this lab you will learn:
1. To know and understand De Morgan’s laws.
2. Know other common Boolean Algebra Theorems
3. Truth Table for De Morgan’s laws and Covering Law
4. Validating De Morgan’s laws, Covering Law, and other Boolean Algebra
De Morgan’s Law
De Morgan’s law states that “AND” and “OR” operations are interchangeable through negation. This law
allows expressing conjunction and disjunction purely in terms of each other through negation.
Conjunction: Conjunction produces a value of true only of both the operands are true. This is commonly
known as AND operator.
Disjunction: Disjunction produces a value of true if either one of the operands are true. This is commonly
known as OR operator.
In simple terms, De Morgan’s laws state that:
Law 1: A NOT OR statement is same as two AND statements with opposite operations.
%%%%%%%%% �  ∪  �  =  �'  ∩ �'
Law 2: A NOT AND statement is same as two OR statements with opposite operations.
%%%%%%%%% �  ∩  �  =  �'  ∪ �'
De Morgan’slaw has been a greattool for digital designersto reduce the size of the digital logic. This
helps
the designers to get uniformity in the design. For example, let’s take the expression %�
%%%+%%%%�
%%�
%%%%
. By
application of De Morgan’s law, this expression reduces to �̅BC. The effective change in circuit is shown
in Figure 1.
∪ = OR
∩ = AND
CMPEN    270    Laboratory    3 2020
2
Figure 1 Circuit size reduction using De Morgan's Law
Other Boolean Algebra Theorems
In class you learned about the following list of boolean algebra theorems. These theorems which will
always be true, will help you when trying to simplify larger equations. Some of them may seem intuitive
but others need a little more proof before believing and understanding them.
Identity: x + 1 = 1
Null element: x + 0 = x
Idempotency: x + x = x
Involution: (x’)’ = x
Compliments: x + x’ = 1
Commutivity: x + y = y + x
Associativity: (x + y) + z = x + (y + z)
Distribution: z(x + y) = zx + zy
Covering: x + xy = x
Combining: xy + xy’ = x
Consensus: xy + x’z + yz = xy + x’z
Demorgan: (xy)’ = x’ + y’
Activities
Activity 1: Truth Table for De Morgan’s Law
As a part of your lab activity, you need to figure out the proper truth table for the De Morgan’s laws.
Please attach the filled out tables in your final report. Take the help of your instructors if you are
having any difficulties. For your reference one row for both the tables (two tables for two laws) are
filled.
CMPEN    270    Laboratory    3 2020
3
A �̅ B �% %
�%%%
∪%%%%
�% �̅∩ �%
0 1 0 1 1
0 1
1 0
1 1
A �̅ B �% %
�%%%
∩%%%%
�% �̅∪ �%
0 1 0 1 1 1
0 1
1 0
1 1
Table 1 De Morgan's laws: Truth table
Activity 2: Verifying the De Morgan’s Law
Once you are done with the truth tables, it’s time to verify the De Morgan’s law in hardware. The circuit
diagram is shown in Figure 3. Instead of using an AND gate, we will use a NAND gate and a NOT gate in
series (Figure 2).
Figure 2 Representing AND gate with NAND gate and NOT gate
Figure 3 De Morgan's Law circuit design
Once you do the connections properly, fill out the table below and attach to your final report. A
format of the table is given below. Also submit a screenshot of your circuit (not waveform).
A B %�%%%∪%%%%�% �̅∩ �% %�%%%∩%%%%�% �̅∪ �%
1 0
1
1
0
0
0
1
0
0
0
0 0
0
0
1
0
0
0
1
0
1
1
0
1
1
0
0 0 1 1
CMPEN    270    Laboratory    3 2020
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Cross verify this table with the table you made above for the truth table and see whether your circuit
works properly, and it verifies De Morgan’s Laws or not.
Activity 3: Truth Table for the Covering Law
Repeat the first two activities to verify the Covering Law: x + xy = x. Start with creating a Truth table
for the Covering Law. Attach a filled out table to your final report.
A B � ∩ � A ∪ (� ∩ �)
0 0
0 1
1 0
1 1
Table 3 Covering law: Truth table
Activity 4: Verifying the Covering Law
You'll notice that this law uses OR and AND. So again, you will have to transform this equation to using
the hardware that we have which are NAND, NOR, and NOT. To perform the OR operation, you will have
to use a NOR then invert the output with a NOT. The same goes for the AND operation, you must
use NAND and then invert the output with a NOT. Submit filled out table to the report, and submit a
screenshot of your circuit (not waveform).
Once you have the hardware connections set up properly use this table to verify the Covering Law.
A B � ∩ � A ∪ (� ∩ �)
Table 4 Verification of Covering Law
Submission Requirements
Once you are finished with all the activities, you are left with 5 filled out tables. That means there should
be 5 tables in your final report that you submit before the due date. There are no questions for this lab
that you must answer, only 5 tables!
2 x Table 1 De Morgan's laws: Truth Table
1 x Table 2 Verification of De Morgan's Law, 1 screenshot of circuit
1 x Table 3 Covering law: Truth Table
1 x Table 4 Verification of Covering Law, 1 screenshot of circuit
1 0
0
0
0
1
0
0
1
1
0 1