made to account for the fact that the set of blue butterflies has been excluded twice. These adjustments, however, are awkward. Boole invented a new system of
“algebra” that avoids such modifications. The axioms
it obeys differ from those of ordinary arithmetic.
The algebra Boole invented proved to be of fundamental importance. It gave 20th-century engineers the
means to instruct machines to follow commands and
has since been used extensively in all computer design
and electrical network theory. In a real sense, Boole
was the world’s first computer scientist, despite the
fact that computers were not invented for another century to come. Boole died unexpectedly in 1864 at the
age of 49 from pneumonia. (The exact date of his
death is not known.)
Boolean algebra In the mid-1800s GEORGE BOOLE
developed a system of algebraic manipulations suitable
for the study of FORMAL LOGIC and SET THEORY, now
called Boolean algebra. He assumed that one is given a
set of elements, which we will denote x, y, z, …, on
which one can perform two operations, today called
Boolean sum, x + y, and Boolean product, x · y. These
operations must satisfy the following rules:
1. The operations are COMMUTATIVE, that is, for all elements x and y we have x + y = y + x and x · y = y · x.
2. There exist two special elements, denoted “0” and
“1,” which, for all elements x, satisfy x + 0 = x and
x · 1 = x.
3. For each element x there is an inverse element “–x”
which satisfies x + (–x) = 1 and x · (–x) = 0.
4. The following DISTRIBUTIVE laws hold for all elements in the set: x · (y + z) = (x · y) + (x · z) and x +
(y · z) = (x + y) · (x + z).
One can see that the Boolean operations “ + ” and “ · ”
are very different from the addition and multiplication
of ordinary arithmetic and so cannot be interpreted as
such. However, thinking of Boolean addition as the
“union of two sets” and Boolean product as “the intersection of two sets,” with 0 being the empty set and 1
the universal set, we see that the all four axioms hold,
making SET THEORY a Boolean algebra. Similarly, the
FORMAL LOGIC of propositional calculus is a Boolean
algebra if one interprets addition as the DISJUNCTION of
two statements (“or”) and product as their CONJUNCTION (“and”).
Other rules for Boolean algebra follow from the
four axioms presented above. For example, one can
show that two ASSOCIATIVE laws hold: x + (y + z) = (x +
y) + z and x · (y · z) = (x · y) · z.
See also DE MORGAN’S LAWS.
Borromean rings The term refers to a set of three
rings linked together as a set, but with the property
that if any single ring is cut, all three rings separate.
The design of three such rings appeared on the coat
of arms of the noble Italian family, Borromeo-Arese.
(Cardinal Carlo Borromeo was canonized in 1610,
and Cardinal Federico Borromeo founded the Ambrosian Art Gallery in Milan, Italy.) The curious
property of the design attracted the attention of
mathematicians.
It is an amusing exercise to arrange four rings such
that, as a set, they are inextricably linked together, yet
cutting any single ring would set all four free. Surprisingly this feat can be accomplished with any number
of rings.
See also KNOT THEORY.
bound A function is bounded if it takes values no
higher than some number M and no lower than some
second value L. For example, the function f(x) = sin x
is bounded between the values –1 and 1. We call M an
upper bound for the function and L a lower bound.
bound 49
Borromean rings
“algebra” that avoids such modifications. The axioms
it obeys differ from those of ordinary arithmetic.
The algebra Boole invented proved to be of fundamental importance. It gave 20th-century engineers the
means to instruct machines to follow commands and
has since been used extensively in all computer design
and electrical network theory. In a real sense, Boole
was the world’s first computer scientist, despite the
fact that computers were not invented for another century to come. Boole died unexpectedly in 1864 at the
age of 49 from pneumonia. (The exact date of his
death is not known.)
Boolean algebra In the mid-1800s GEORGE BOOLE
developed a system of algebraic manipulations suitable
for the study of FORMAL LOGIC and SET THEORY, now
called Boolean algebra. He assumed that one is given a
set of elements, which we will denote x, y, z, …, on
which one can perform two operations, today called
Boolean sum, x + y, and Boolean product, x · y. These
operations must satisfy the following rules:
1. The operations are COMMUTATIVE, that is, for all elements x and y we have x + y = y + x and x · y = y · x.
2. There exist two special elements, denoted “0” and
“1,” which, for all elements x, satisfy x + 0 = x and
x · 1 = x.
3. For each element x there is an inverse element “–x”
which satisfies x + (–x) = 1 and x · (–x) = 0.
4. The following DISTRIBUTIVE laws hold for all elements in the set: x · (y + z) = (x · y) + (x · z) and x +
(y · z) = (x + y) · (x + z).
One can see that the Boolean operations “ + ” and “ · ”
are very different from the addition and multiplication
of ordinary arithmetic and so cannot be interpreted as
such. However, thinking of Boolean addition as the
“union of two sets” and Boolean product as “the intersection of two sets,” with 0 being the empty set and 1
the universal set, we see that the all four axioms hold,
making SET THEORY a Boolean algebra. Similarly, the
FORMAL LOGIC of propositional calculus is a Boolean
algebra if one interprets addition as the DISJUNCTION of
two statements (“or”) and product as their CONJUNCTION (“and”).
Other rules for Boolean algebra follow from the
four axioms presented above. For example, one can
show that two ASSOCIATIVE laws hold: x + (y + z) = (x +
y) + z and x · (y · z) = (x · y) · z.
See also DE MORGAN’S LAWS.
Borromean rings The term refers to a set of three
rings linked together as a set, but with the property
that if any single ring is cut, all three rings separate.
The design of three such rings appeared on the coat
of arms of the noble Italian family, Borromeo-Arese.
(Cardinal Carlo Borromeo was canonized in 1610,
and Cardinal Federico Borromeo founded the Ambrosian Art Gallery in Milan, Italy.) The curious
property of the design attracted the attention of
mathematicians.
It is an amusing exercise to arrange four rings such
that, as a set, they are inextricably linked together, yet
cutting any single ring would set all four free. Surprisingly this feat can be accomplished with any number
of rings.
See also KNOT THEORY.
bound A function is bounded if it takes values no
higher than some number M and no lower than some
second value L. For example, the function f(x) = sin x
is bounded between the values –1 and 1. We call M an
upper bound for the function and L a lower bound.
bound 49
Borromean rings