Sets

Definition: A well defined collection of objects.

Elements or members: The individual objects in a set.

We generally use capital Latin letters to name sets and lowercase letters to denote elements in a set. If the element x is an element of the set A we use the notion xÎA (read: x belongs to set A).     

Drawings called Venn diagrams are used to visualize relationships among sets.

Cardinal number: The number of elements in set A is called the cardinal number (absolute value) of  set A and is denoted by n(A) or│A│.

A set is finite if it’s cardinal number is is a whole number. An infinite set is one that is not finite.

Cartesian product: Let A and B two non-empty sets. The Cartesian product of  two sets A and B, denoted by AxB, consists of the set of all ordered pairs (a;b) such that  a is belong to A and b is belong to B. AxB={(a;b)│     .aÎA and bÎB}

Special sets:

Empty or null set: The set that contains no elements. This set is labelled by the symbol Ø.

Equal sets: Two sets A and B are equal if they have exactly the same members.In this cas we write A = B.

Subset: The set A is a subset of the set B if every element of set A is also an element of set B.

In order to show that A is included in B , we must show that every element of A also occures as an element of  B.To show that A is not a subset of B, we have to fid only one element of A that is not in B.

A set has n elements has 2ⁿ subsets.

Set operations

Union: The union of sets A and b, written as AUB is the set of elements that are members of either A or B or both.

Intersection: The intersection of sets A and B, written as A∩B, is the set of elementscommon to both A and B.

Disjoint: The intersection of more than two sets is the set of elements that belong to each of the sets. If AÇB= Ø, then we say A and B  are disjoint.

Complement: If A is a subset of the universal set U, the complement of A is the set of elemnets of U that are not elements os A. This is denoted by A’.

Difference: The difference of sets A and B is the set of elements that are in B but not in A. This set is denoted by B\A or B – A.

Number sets

C = natural numbers ={1, 2, 3, …}

W = whole numbers ={0, 1, 2, 3…}

I = integers ={…-2, -1, 0, 1, 2, 3….}

Q = rational numbers = {x is the form of  ^a/_b, where a and b are integers and b is not equal to 0}

Q* = irrational numbers = { x is not a rational number}

Number sets in Hungary

N = natural numbers ={0, 1, 2, 3, …}

Z = integers ={…-2, -1, 0, 1, 2, 3….}

Q = rational numbers = {x is the form of  ^a/_b,  where a and b are integers and b is not equal to 0}

Q* = irrational numbers = { x is not a rational number}

R = real numbers {every existing number}

Intervals

Closed interval: interval from a to b. Denotion:                   [a;b]                a £ x £ b

Open interval: interval from a to b. Denotion:                   (a;b)                  a< x < b

Interval from a to b  open from the left . Denotion:             (a;b]                 a £ x < b

Interval from a to b open from the right . Denotion:           [a;b)                 a < x £ b

Number half line from a to + ¥ . Denotion:                             [a;+ ¥)            a £ x

Open Number half line from a to + ¥. Denotion:                     (a; ¥)               a

Number half line from a to - ¥.   Denotion:                             (-¥;a]              x£ a

Open number half line from – to a  ¥. Denotion:                     (-¥;a)              x