pairwise disjoint (independent, mutually exclusive)
A collection of sets A, B, C, … is said to be pairwise
disjoint if the intersection of any two sets from the collection is empty. For example, the sets {1,2}, {3,4},
{5,6}, … are pairwise disjoint, as are the three sets:
A = the set of all even natural numbers
B = the set of all natural numbers 1 greater than a
power of 100
C = the set of all prime numbers 1 less than a multiple
of 4
A collection of pairwise disjoint subsets A, B, C,
… of a larger set S is said to be exhaustive if every element of S is listed as an element of (just one) specified
subset. In this case we also say that the sets A, B, C, …
partition the set S. The first example presented above,
for instance, is an exhaustive subset of the natural
numbers and so partitions the positive whole numbers.
The second example, however, does not. (The number
13, for instance, is not a member of any of the specified subsets.)
Given a partition A, B, C, … of a set S, two elements a and b of S are said to be equivalent, with
respect to that partition, denoted a ~ b, if they belong
to the same subset specified by the partition. For example, the days of the year are partitioned by seven disjoint sets given by the weekday names of the days. For
instance, August 1, 1966, and June 30, 2003, are
equivalent in this context since they both belong to the
subset called “Monday.”
The notion of equivalence satisfies three key
properties:
i. Reflexivity: any element a is equivalent to itself: a ~ a
ii. Symmetry: if a is equivalent to b, then b is equivalent to a: a ~ b ⇒ b ~ a
iii. Transitivity: if a is equivalent to b and b is equivalent
to c, then a is equivalent to c: a ~ b, b ~ c ⇒ a ~ c
In general, any relationship “~” defined on elements of a set S satisfying the three properties listed
above is called an equivalence relation on S. For example, deeming two words of the English language to be
equivalent if they each possess the same number of
vowels is an equivalence relation on the set of all
words. It turns out that any equivalent relation, no
matter how it is defined, arises from a partition of the
set on which it is based. For example, the set of all
words is partitioned by the sets W 0 ,W 1 ,W 2 , … where
W k is the set of all words with precisely k vowels.
See also DAYS-OF-THE-WEEK FORMULA; SET THEORY.
Pappus of Alexandria (ca. 300–350 C.E.) Greek Geometry Born in Alexandria, Egypt, Pappus is considered
today to be the last great geometer of antiquity to work
in the Greek way of thought and scholarly tradition. He
wrote commentaries on the works of EUCLID and
CLAUDIUS PTOLEMY, but is most notably remembered
for his treatise Synagoge (Collections), of which volumes
III–VII of the original eight have survived intact today.
371
P
A collection of sets A, B, C, … is said to be pairwise
disjoint if the intersection of any two sets from the collection is empty. For example, the sets {1,2}, {3,4},
{5,6}, … are pairwise disjoint, as are the three sets:
A = the set of all even natural numbers
B = the set of all natural numbers 1 greater than a
power of 100
C = the set of all prime numbers 1 less than a multiple
of 4
A collection of pairwise disjoint subsets A, B, C,
… of a larger set S is said to be exhaustive if every element of S is listed as an element of (just one) specified
subset. In this case we also say that the sets A, B, C, …
partition the set S. The first example presented above,
for instance, is an exhaustive subset of the natural
numbers and so partitions the positive whole numbers.
The second example, however, does not. (The number
13, for instance, is not a member of any of the specified subsets.)
Given a partition A, B, C, … of a set S, two elements a and b of S are said to be equivalent, with
respect to that partition, denoted a ~ b, if they belong
to the same subset specified by the partition. For example, the days of the year are partitioned by seven disjoint sets given by the weekday names of the days. For
instance, August 1, 1966, and June 30, 2003, are
equivalent in this context since they both belong to the
subset called “Monday.”
The notion of equivalence satisfies three key
properties:
i. Reflexivity: any element a is equivalent to itself: a ~ a
ii. Symmetry: if a is equivalent to b, then b is equivalent to a: a ~ b ⇒ b ~ a
iii. Transitivity: if a is equivalent to b and b is equivalent
to c, then a is equivalent to c: a ~ b, b ~ c ⇒ a ~ c
In general, any relationship “~” defined on elements of a set S satisfying the three properties listed
above is called an equivalence relation on S. For example, deeming two words of the English language to be
equivalent if they each possess the same number of
vowels is an equivalence relation on the set of all
words. It turns out that any equivalent relation, no
matter how it is defined, arises from a partition of the
set on which it is based. For example, the set of all
words is partitioned by the sets W 0 ,W 1 ,W 2 , … where
W k is the set of all words with precisely k vowels.
See also DAYS-OF-THE-WEEK FORMULA; SET THEORY.
Pappus of Alexandria (ca. 300–350 C.E.) Greek Geometry Born in Alexandria, Egypt, Pappus is considered
today to be the last great geometer of antiquity to work
in the Greek way of thought and scholarly tradition. He
wrote commentaries on the works of EUCLID and
CLAUDIUS PTOLEMY, but is most notably remembered
for his treatise Synagoge (Collections), of which volumes
III–VII of the original eight have survived intact today.
371
P