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Introduction | Terminology | References

Introduction

Web site "Set Theory" offers educational tools and resources to help you learn fundaments of mathematics. You can take full advantage of the "Internet style of learning."

Set theory was developed in the second half of the Nineteenth Century. It has its roots in the work of Georg Cantor, although contributions of others were significant. Ultimately, the goal of Set Theory was to provide a common axiomatic basis for all of mathematics.

Here you can find out something about sets, basic operations on sets, functions, relations, Cartesian product, denumerable sets and cardinality.

If we consder that the team responseable for this site is one set, then its members are: Tina Skrtic, Matej Loncaric and Sandra Basic(coach).

TERMINOLOGY AND NOTATION

 ALEPH aleph is used for the Hebrew letter aleph ALEPH ZERO 0 aleph0 is used for aleph-zero, the zero'th well-ordered infinite cardinal; aleph0=w = {0, 1, 2, ...} BIJECTION A function is bijective or a bijection or a one-to-one correspondence if it is both injective and surjective. There is exactly one element of the domain which maps to each element of the co-domain. CARDINAL NUMBER k(A) or card(A) A cardinal number is one way to measure the size of a set. CARDINALITY The cardinal (or cardinality) of x, denoted k(x), is: the least ordinal a equinumerous to x, if x is well-orderable, and the set of all sets y of least rank which are equinumerous to x, otherwise. CARDINALLY EQUIVALENT Two sets A and B are said to be cardinally equivalent, if and only if there exists a bijection between them. CARTESIAN COORDINATE A pair of numbers, (x, y), defining the position of a point in a two-dimensional space by its perpendicular projection onto two axes which are at right angles to each other. x and y are also known as the abscissa and ordinate. CARTESIAN PRODUCT The Cartesian product of two sets A and B is the set A x B = {(a, b) | a in A, b in B}.I.e. the product set contains all possible combinations of one element from each set. COMPLEMENT AC AC ={x | xU and xA}. COMPOSITION g ° f If f: X  Y and g: Y  Z, then you can get from X to Z by combining f and g into a single function g ° f. g ° f is defined to be x | g(f(x)). CONSTANT FUNCTION A function with no arguments, or, one which always gives the same value. CO-DOMAIN The set of values containing all possible results of a function. The codomain of a function f of type D -> C is C. A function's image is a subset of its codomain. DENUMERABLE SETS A set A which is cardinally equivalent to the set of natural numbers is called denumerable or countably infinite. DIFFERENCE The difference or relative comlement of A: B-A or B\A={x | xB and xA}. DISJOINT SETS A set whose members do not overlap, are not duplicated, etc. DOMAIN The set of argument values for which a function is defined. EACH ELEMENT Whatever belongs to a set are its elements. NOT ELEMENT EQUALITY OF SETS A=B Set A is equal to set B if they both have the same members. NOT EQUAL EQUIVALENCE RELATION ~ If a relation is reflexive, symmetric, and transitive, it is called an equivalence relation on the set A. EXIST FINITE SET A set A is finite if and only if either A is empty or, for some nN , the set {1, 2, 3, … n}~A. FUNCTION f :XY If X and Y are sets, then f is a function from X to Y whenever f assigns a unique element of Y to each element of X. X is the domain of f, and Y is the co-domain.f(x) is assigned to x, while x is mapped to f(x). IDENTITY FUNCTION 1E A function from a set onto itself, which leaves every element of the set unchanged. IMPLIES INFINITE SET A set with an infinite number of elements. There are several possible definitions, e.g. A set X is infinite if there exists a bijection (one-to-one mapping) between X and some proper subset of X. A set X is infinite if there exists an injection from the set of natural numbers N to X. INJECTION A function, f : A  B, is injective or one-one, or is an injection, if and only if for all a,b in A, f(a) = f(b) = > a = b. < /STRONG > INTERSECTION The intersection of A and B : AB={x | xA or xB}. INVERSE f-1 Let f : D  C, a function g : C  D is called a left inverse for f if for all d in D, g (f d) = d and a right inverse if, for all c in C, f (g c) = c and an inverse if both conditions hold. Only an injection has a left inverse, only a surjection has a right inverse and only a bijection has inverses. The inverse of f is often written as f with a -1 superscript. INVERSE OF A RELATION R-1 R-1={(x,y) | (y,x)R NULL SET ORDERED PAIR (x,y) Set of two numbers in which the order has an agreed-upon meaning, such as the Cartesian coordinates (x, y), where the first coordinate represents the horizontal position, and the second coordinate represents the vertical position. ORDERED SET A set A is called ordered if it is partially ordered and every pair of elements x and y from the set A can be compared with each other via the partial ordering relation. PARTITION P A division of a set into nonempty disjoint sets which completely cover the set POWER SET 2A The set of all possible subsets of a given set A is called the power set of A. RELATIONS A set of ordered pairs is called a relation. SET A set is a collection of things of any kind. If B is a set we call the "things" in B the elements or members of B. In symbols bB means that b is an element of B. Similarly, for a set B the statement bB means that the object b is not in B. Sets themselves are often symbolized by enclosing their elements within "curly brackets" {}. Sets can also be described by a rule members must satisfy. If P(x) is the rule "x is beautiful Blondie", then the set listed above could be symbolized as {x | P(x) }. We say that "the set of all x such that P(x)". The "|" stands for the expression "such that". The use of curly brackets and a rule to specify sets is called set-builder notation and is used throughout mathematics SUBSET Subset is a set which is part of another set. If X and Y are sets, then Y is a subset of X if and only if, for all possible elements x, x is an element of X whenever x is an element of Y. PROPER SUBSET SUPERSET NOT SUBSET SURJECTION A function f : A  B is surjective or onto or a surjection if f (A )= B. I.e. f can return any value in B. This means that its image is its co-domain. < /FONT> < /STRONG> UNION The union of A and B : AB={x | xA and xB}. UNIVERSE U We consider that all sets are subsets of some finite universe U. Venn diagram The rectangle represents universe U and any subset of U is drawn as a closed figure within this rectangle.

References

Books:

1. Kurepa, Svetozar. Uvod u matematiku. Zagreb, 1975.

2. Blanusa, Danilo. Matematika I. Zagreb, 1980.

3. Devide, Vladimir. Zadaci iz apstraktne algebre. Zagreb, 1981.

4. Lipschutz, Seymour. Set Theory and Related Topics (Schaum's Outline Series). New York, 1964.

5. Sporer, Zlatko. Repetitorij matematike za osnovnu skolu. Zagreb, 1992.

Electronic sources:

1. An elementary Introduction to Logic and Set Theory, http://www.madison.tec.wi.us/alehnen/weblogic/logset.htm,

Last visited: August, 2001