DeMorgan's theorems state that: 1) The negation of a conjunction is the disjunction of the negations. 2) The negation of a disjunction is the conjunction of the negations. They can be used to transform logical expressions between conjunctive and disjunctive normal form by distributing negations inward and changing conjunctions to disjunctions (or vice versa). The document provides examples of applying DeMorgan's theorems to logically equivalent expressions.

1. 1
Forging new generations of engineers

2. 2
DeMorgan’s Theorems

3. 3
DeMorgan’s Theorem #1
A · B = A + B
EQUAL
A B A x B A x B A B A + B
0 0 0 1 0 0 1
0 1 0 1 0 1 1
1 0 0 1 1 0 1
1 1 1 0 1 1 0

4. 4
DeMorgan’s Theorem #2
A B A + B A + B A B A x B
0 0 0 1 0 0 1
0 1 1 0 0 1 0
1 0 1 0 1 0 0
1 1 1 0 1 1 0
A + B = A · B
EQUAL

5. 5
DeMorgan’s Theorem #1
A · B = A + B
B
A
B
A
=
Invert output of an
AND gate
Invert the inputs of an
OR gate

6. 6
DeMorgan’s Theorem #2
A + B = A · B
=
Invert output of an
OR gate
Invert the inputs of an
AND gate
B
A
B
A

7. 7
DeMorganizing an Expression
XCBCACBCA =))(+(+)(+)(
DeMorg. XCBCACBCA =)))(+(())((
DeMorg. XCBCACBCA =)))(+()(+)((
Dist. XCBCACACBCA =)))(+()(+(
XCBCACBCA =)))(+()()((AA =

8. 8
DeMorganizing an Expression
XCBCACACBCA =)))(+()(+(
0=AA XCBCAABCA =)))(+()(0+(
0=0A XCBCABCA =)))(+()(0+(
DeMorg
XCBCABCA =))(+)+)(((
XCBCABCA =)))(+()((AA =0+

9. 9
DeMorganizing an Expression
XCBCABCA =))(+)+)(((
DeMorg XCBCABCA =))+(+))(((
Assoc. XCBCABCA =)++)((
Dist. XCBCABBCACABCA =++
0=0A XBABBCACBC =0++0
XBBCA =0++0AA =0+

10. 10
DeMorganizing an Expression
XBBCA =0++0
AA =0+ XBBCA =
XBCA =AAA =