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Demorgan's Theorem

DeMorgan's Theorems are two fundamental rules in Boolean Algebra that explain the relationship between NOT, AND, and OR operations, essential for simplifying logic expressions in digital circuit design. The first theorem states that the complement of an AND operation is equivalent to the OR of the complements, while the second theorem states that the complement of an OR operation is equivalent to the AND of the complements. These theorems are crucial for optimizing digital systems and are widely used in basic electronics engineering.

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57 views9 pages

Demorgan's Theorem

DeMorgan's Theorems are two fundamental rules in Boolean Algebra that explain the relationship between NOT, AND, and OR operations, essential for simplifying logic expressions in digital circuit design. The first theorem states that the complement of an AND operation is equivalent to the OR of the complements, while the second theorem states that the complement of an OR operation is equivalent to the AND of the complements. These theorems are crucial for optimizing digital systems and are widely used in basic electronics engineering.

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aslimadhav146
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Explain Demorgans theom with Truth

values andBoolean expression in


basic electronics engineering...

Certainly! Here's a detailed explanation of


DeMorgan's Theorems - including Boolean
expressions, truth tables, and their use in Basic
Electronics Engineering (FY B.Tech).

What is DeMorgan's Theorem?


DeMorgan's Theorems are two important rules in
Boolean Algebra that describe how NOT (inversion)
operates over AND and OR operations.

They are extremely useful in simplifying logic


expressions and are the foundation for designing
digitalcircuits using universal gates like NAND and
NOR.
DeMorgan's Theorems - Boolean
Expressions
There are two theorems, also called DeMorgan's
Laws:

DeMorgan's First Theorem:

(A- B) = A + B'
The complement of the AND operation is equal to
the OR of the complements.

DeMorgan's Second Theorem:

(A + B)=A B'
The complement of the OR operation is equal to
the AND of the complements.
Truth Tables for Both Theorems

First Theorem: (A. BY = A + B

AB A·B (AB)' A' B' A'+B'


0 0 1 1 1 1

010 1 1 01

1 0 0 1 0 11
11 1 0 0

Result: Column (A-B)' matches A'+B'


Hence, first theorem is verified
| Second Theorem: (A + B) = A'B

AB A+B (A+B)' A'B' A-B'


0 0 0 1 1 1 1

0 11 0 1 0

1 01 0 010
111 0 0

Result: Column (A+B)' matches A'·B'


Hence, second theorem is also verified
Logic Interpretation Using Gates
First Theorem: (A · B) = A' + B
Left side:

" AND gate then NOT (NAND)

Right side:

" NOT A and NOT B then OR

Both give the same result


Second Theorem: (A + B) = A-B'
Left side:

" OR gate ’ then NOT (NOR)

Right side:

" NOT Aand NOTB then AND

Both give the same result


L Importance in Basic Electronics:
"Essential for simplifying logic circuits
" Used in Karnaugh Map reduction
"Allows building all logic gates using only NAND
or NOR gates
" Helps in optimizing digital system design
" Common in programmable logic devices and
embedded circuits

Summary Table

Theorem No. Expression Meaning


1st (A- BY =A+ B' NOT of ANI
2nd (A+ B)=AB NOT of OR
Summary Table

Expression Meaning

(A·B)= A + B NOT of AND is OR of NTS


(A+ B) =A·B' NOT of OR is AND of NOTS
Final Note:

DeMorgan's Theorems help you:

"Understand how logic gates work


Design digital circuits using fewer components
" Simplify expressions in Boolean algebra
" Convert expressions from one logic family to
another (like NOR-based design)

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