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Basic notions. Examples of sets
Set.
Element of a set. Finite set. Empty set.
Infinite set. Countable set.
Uncountable set.
Convex set. Methods of
description of sets.
A set
and an element of a set
concern with category of primary notions, for which it's impossible to
formulate the strict definitions. So, we imply as sets usually collections of
objects (
elements of a set
), having certain common properties.
For instance, a set of books in a
library, a set of cars on a parking lot, a set of stars in the sky, a world of
plants, a world of animals – these are examples of sets.
A finite set
consists of finite number of
elements, for example,
a set of pages in a book, a set of
pupils in a school etc.
An empty set
( its
designation is  )
doesn't contain any elements, for instance, the set of winged elephants,
the set of roots of the equation sin x = 2 etc.
An infinite set
consists of infinite number of
elements, i.e. this is a set, which isn't finite and empty.
Examples: the set
of real numbers,
a set of points
on a plane,
a set of atoms in the
universe etc.
A countable set
is a set, elements
of which can be numbered.
For example,
the sets of natural, even,
odd numbers. A countable set can be finite ( a set of books in a library ) or
infinite ( the set of integers, its elements can be numbered as follows:
the set elements:
…, –5, – 4, –3, –2, –1, 0, 1, 2, 3, 4, 5, …
their numbers:
… 11 9 7
5 3 1 2 4 6 8 10 … ) .
An uncountable set
is a set, elements of which
can't be numbered.
For
example,
the set of real numbers. An uncountable set can be only infinite (
think,
please,
why
?
).
A convex set
is a set, which for any two its
points
A
and B contains also the whole segment
AB.
Examples of convex sets: a straight line, a plane, a circle. But a circumference
is not a convex set.
Methods of description of sets.
A set can be described
the following ways:
– an enumeration of all its elements
by theirs names (
for example, a set of books in a
library, a set of pupils in a class, an alphabet of any language and so
on
);
– by giving
of common performance (common
properties) of elements of the set (
for
instance,
the
set
of rational
numbers,
the
family
of
dogs,
the
family
of
cats
etc.);
– formal law of forming elements of
the set (
for example, the formula of a
general term of numerical sequence, Periodic table of chemical elements
).
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