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 ).