Activity 2.1.5 DeMorgan’s Theorems
Introduction
Despite all of the work done by George Boole, there was still more work to be done.
Expanding on Boole’s studies, Augustus DeMorgan (1806-1871) developed two additional
theorems that now bear his name. Without DeMorgan’s Theorems, the complete
simplification of logic expression would not be possible.
Theorem #1:
XY  X Y
Theorem #2 :
X Y  XY
As we will seen in later activities, DeMorgan’s Theorems are the foundation for the NAND
and NOR logic gates.
In this activity you will learn how to simplify logic expressions and digital logic circuits using
DeMorgan’s two theorems along with the other laws of Boolean algebra.
Procedure
Using DeMorgan’s theorems and the other theorems and laws of Boolean algebra, simplify
the following logic expressions. Note the theorem/law used at each simplification step. Be
sure to put your answer in Sum-Of-Products (SOP) form.
1.
F1  X  Y
F= X*Y’
2.
F2  ( R  S ) ( T  U )
F= (R’+S)+(T+U’)
3.
F3  M N
F= M*N’
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Digital Electronics Activity 2.1.5 DeMorgan’s Theorems – Page 1
4.
F4  W X ( Y  Z )
F= WX’+Y+Z’
5. F5  P Q R  P Q R
F= (P’+Q+R’)(P+Q’+R)
6. F6  W X Y Z
F= W+X’+Y’+Z
Let’s see how we would utilize DeMorgan’s theorems to simplify a digital logic circuit.
7. Write the UN-SIMPLIFIED logic expression for the output Do-Nothing in the logic
circuit shown below.
F=(X’+Y”)(Y’Z’”)+(Y’+X”)
The output is call Do-Nothing because that is exactly what the circuit does, nothing; yet it’s a
good example for learning about DeMorgan’s theorems.
8. Using DeMorgan’s theorems and the other theorems and laws of Boolean algebra,
simplify the logic expression Do Nothing. Be sure to put your answer in Sum-OfProducts (SOP) form.
F=X’+X’YZ’+Y’Z
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Digital Electronics Activity 2.1.5 DeMorgan’s Theorems – Page 2
9. In the space provided, draw an AOI circuit that implements the simplified logic
expression Do Nothing. For this implementation you may assume that AND & OR
gates are available with any number of inputs.
VCC
Do_Nothing
5V
2.5 V
U1
X
U5 NOT
Key = Space
AND2
Y
U2
U4
AND2
U6
NOT
U8
OR2
OR2
U7
Key = Space
U3
Z
AND3
NOT
GND
Key = Space
Do Nothing – I
10. Re-implement the circuit assuming that only 2-input AND gates (74LS08), 2-input OR
gates (74LS32), and inverters (74LS04) are available. Draw this circuit in the space
provided.
VCC
5V
X_
Key = Space
Y_
U9
Do__Nothing
U13NOT
U12
AND2
U10
AND2
2.5 V
U14
U15
NOT
U17
U11
AND2
Key = Space
Z_
NOT
GND
OR2
U16
OR2
AND2
Key = Space
Do Nothing – II
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Digital Electronics Activity 2.1.5 DeMorgan’s Theorems – Page 3
Conclusion
1. Draw the gate equivalent for DeMorgan’s two theorems.
2.
Theorem #1:
XY  X Y
U15
OR2
Theorem #2 :
X Y  XY
U16
AND2
3. How would you prove that the original Do Nothing circuit and the simplified version
are equivalent?
By making a truth table and comparing them.
4. If each GATE cost 5¢ and you made 100,000 of the unsimplified units, how much of
the company’s money did you waste on the unsimplified Do-Nothing project?
25,000.oo
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Digital Electronics Activity 2.1.5 DeMorgan’s Theorems – Page 4