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Augustus
DeMorgan was one of the most brilliant mathematicians of his time.
In 1826, he returned to his home in London. In 1827, at the age of
21, he applied for the chair of mathematics in the newly founded
University College London, and despite having no mathematical
publications he was appointed. In 1828 DeMorgan became the first
professor of mathematics at University College and a founder of
the London Mathematical Society. He gave his inaugural lecture on
the study of mathematics. DeMorgan resigned his chair, on a matter
of principle, in 1831. He was appointed to the chair again in
1836, and held it until 1866.
In
1838 he defined and introduced the term 'mathematical induction',
putting a process that had been used without clarity. He
recognized the purely symbolic nature of algebra and he was aware
of the existence of algebras other than ordinary algebra. He
introduced DeMorgan's Laws and his greatest contribution is as a
reformer of mathematical logic.
He was the one that came up with the idea of using a slash
to represent fractions, as in 1/5 and 3/7. In addition, he
perfected the principle of Mathematical
Induction and gave it its name in 1838. Today, the
principle of Mathematical Induction is one of the most widely used
methods of proof for proving statements and arguments.
However,
DeMorgan would be best remembered for his many contributions to
the development of abstract symbolic logic; he and George
Boole are regarded as the founders of this relatively
new branch of Mathematics. His DeMorgan’s laws, which were named
after him, stated:
"The
negation of an "and" statement is logically equivalent
to the "or" statement in which each component is
negated.
"The negation of an "or" statement is logically
equivalent to the "and" statement in which each
component is negated."
DeMorgan’s laws
were an important element in the field of symbolic logic. It made
things easier for other mathematicians to study symbolic logic and
made them more receptive towards this abstract and radical study.
DeMorgan’s laws also played an important role in the study of sets
theory.
In
1866 he was a co-founder of the London Mathematical Society and
became its first president. DeMorgan's son George, a very able
mathematician, became its first secretary. In the same year, De
Morgan was elected a Fellow of the Royal Astronomical Society.
DeMorgan's
Theorem
1. (xy)'
= x' + y'
2. (x +
y)' = x'y'
To
apply DeMorgan's Theorem,
- compliment each variable
- replace each operator with its dual, AND
with OR, OR with AND
You
can use DeMorgan's to change an SOP to a POS
xy =
(x' + y')'
You
can use DeMorgan's to change a POS to an SOP
x + y =
(x'y')'
Let's
look at the Exclusive Or function
ab | f
g minterms maxterms
---|----
--- ---
00 | 0
1 a'b' a'+b'
01 | 1
0 a'b a'+b
10 | 1
0 ab' a+b'
11 | 0
1 ab a+b
remember:
the SOP of the minterms = POS of the maxterms
f =
a'b+ab' = (a'+b')(a+b)
this
can be shown by multiplying the POS
(a'+b')(a+b)
= a'a+a'b+b'a+b'b = a'b+ab' QED
now
compliment the POS for f and apply DeMorgan's
f, POS
= (a'+b')(a+b)
f' = [(a'+b')(a+b)]'
= (a'+b')'+(a+b)' = ab+a'b'
and
write the SOP for g
g, SOP
= a'b'+ab
therefore
g = f'
Examples
of applying DeMorgan's
POS
--> SOP
(a+b)(c+d)
= (a+b)' + (c+d)' = a'b'
+ c'd'
SOP
--> POS
(ab'+c'd+ef)'
= (ab')'(c'd)'(ef)' = (a'+b)(c+d')(e'+f')
MIXED
(w+x+y)z
= (w+x+y)' + z' = w'x'y'
+ z'
(a+b)cd
= (a+b)'+c'+d' = a'b'+c'+d'
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