Beginning Mathematics/Set Theory

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In this section we shall go over some basic set theory.

Contents

[edit] Sets

A set is a well defined collection of objects. a collection is said to be well defined,if we can determine whether the given object belongs to the collection or not. the objects of the set are called elements. we note that the elements may not be numbers. Elements of sets may be letters, objects, or even other sets. Sets can be finite, or infinite.Sets are denoted by captial letters of english such as A,W,D,C,V,X,Z.. . The elements of the set are indicated between the flower braces,usually with small letters of english, seperated by commas. Ex: A={2,4,6,8,10}. Because all the elements are listed it is called List form. This is also called Roster form. Often, sets will have a pattern of elements or the elements will be similar in some fashion. At times, when it is clear what the pattern is, the set will end with "..." indicating the elements continue in that fashion. Ex:X={1,3,5,7,9,11,...}. Other times the set may be described in words or mathematical notation. Mathematical notation is read as in the following example: for the set X X={1,3,5,7,9,11,...}. This is shown as X={x/x is an odd number}

we read this as X is a set of all x's such that x is an odd number.  In general, this is read as " X is the set of x such that x has a property".
Here the / stands for "such that" often a colon (:) will be used instead.

To denote an element is in a set we write: x \in A this reads "x is an element of the set A" or "x is in A".

[edit] Special Sets

Some sets are used frequently in mathematics so they are given by special notation.

\mathbb{N} = \mathcal{f}Set\;of\;whole\;numbers\mathcal{g} = \{0,1,2,3,4,5,6,...\}

\mathbb{Z} = \{Set\;of\;all\;integers \} = \{...,-3,-2,-1,0,1,2,3,...\}

\mathbb{Q} = \{Set\;of\;rational\;numbers\} = \{\frac{a}{b} \; \mathcal{j} \; a,b \in \mathbb{Z} \; and \; b \neq 0\}

\mathbb{R} = \{Set\;of\;real\;numbers\}

\mathbb{C} = \{Set\;of\;complex\;numbers\} = \{a + bi \; \mathcal{j} \; a,b \in \mathbb{R} \; and \; here \; i^2=-1 \}

[edit] Examples

\mathcal{f}1,2,3 \}

\mathcal{f}7,12,80\}

\{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}= \{letters\; of\; english\; abc\}

\mathcal{f}1,2,3,4,5,...\} = \{Set\; of\; positive\; integers\}

\{2,4,6,8,10,12,14...\} = \{ Set\; of\; positive\; even\; integers \} = \{x | x \in \mathbb{Z}\; and\; x\; is\; even \}

[edit] Basic operations

[edit] Union

The union of two sets is denoted by \bigcup . The union of two sets is best described as "everything in both the sets once".

Formally: A \bigcup B = \{x | x \in A \; or \; x \in B\}

For example:

A = \mathcal{f}1,2,3\} B = \mathcal{f}4,5,6\}

A \bigcup B = \{1,2,3\} \bigcup \{4,5,6\} = \{1,2,3,4,5,6\}

[edit] Intersection

The intersection of two sets is denoted by \bigcap . The intersection of two sets is best described as "only what is in both of the sets".

Formally: A \bigcap B = \{x | x \in A \; and \; x \in B\}

For example:

A = \mathcal{f}1,2,3,4,5,6\} B = \mathcal{f}2,4,6,8,10\}

A \bigcap B = \{1,2,3,4,5,6\} \bigcap \{2,4,6,8,10\} = \{2,4,6\}

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