04Asetnotation

= Set Notation =


 * Definitions **


 * A ** set ** is a collection of things (objects, values, people, events, ...)
 * Each thing inside a set is called an ** element ** of that set.


 * To list the contents of a set, we use curly brackets { } and a comma between each element.
 * We name each set with a capital letter
 * A = { 1, 3, 5, 7 }
 * B = { 1, 2, 3, 4 }


 * Venn Diagrams **


 * A ** Venn Diagram ** is a way of visually representing a set
 * A set is represented by a circle
 * Elements of a set are shown inside the circle
 * The rectangle represents the Universal Set ( e )
 * The Universal Set ( e ) contains everything being considered


 * The symbol Î means "is an element of the set"
 * 5 Î A
 * 1 Î B
 * The symbol Ï means "is __not__ an element of the set"
 * 2 Ï A


 * The symbol Ì means "is a subset of the set" ..
 * a subset is a smaller set completely contained within the main set
 * { 1, 5 } Ì A
 * { 4 } Ì B
 * The symbol Ë means "is __not__ a subset of the set"
 * { 6, 7, 8 } Ë A


 * Union of two Sets **


 * The Union means the set of all elements that belong to either of the two sets,
 * It is the joining together of all the different elements in both sets
 * in the same way that a workers union is the joining together of
 * all the different workers from that type of industry.
 * The Union of A and B is shown shaded in green.
 * The symbol È means "the Union of the two sets"
 * A È B = { 1, 2, 3, 4, 5, 7 }
 * { a, b, c } È { c, d, e } = { a, b, c, d, e }


 * The Intersection of two Sets **


 * The Intersection means the set of elements that appears in both of the two sets
 * It is the place where the two sets __overlap__ in the same way that
 * the intersection between two roads is the place where those
 * roads overlap
 * The Intersection of A and B is shown shaded in green
 * The symbol Ç means "the Intersection of the two sets"
 * A Ç B = { 1, 3 }
 * { a, b, c } Ç { c, d, e } = { c }


 * The Null Set **


 * The symbol Æ means the Null Set (or the Empty Set)
 * The Null Set has no elements
 * Æ = { }


 * The Complement of a Set **


 * The __complement__ of a set is everything __outside__ of the set
 * The complement of A is shown shaded in green
 * The symbol A ' means "the complement of A"
 * A ' = { 2, 4, 6, 8 }


 * Excluding values **


 * The symbol A \ B denotes all of the elements of A **__excluding__** those in B
 * A \ {1} = {3, 5, 7}
 * {1, 2, 3} \ {3, 4, 5} = {1, 2}
 * R \ {1,2} means all real numbers __except__ 1 and 2


 * Sets of Numbers **

Particular letters are reserved as definitions for the following sets of numbers.




 * R = the set of Real Numbers
 * this includes all numbers that can be shown on a number line
 * R is divided into two subsets
 * Q = Rational Numbers
 * Q ' = Irrational Numbers


 * Q = the set of Rational Numbers
 * this includes all numbers that can be expressed as a fraction
 * Q is divided into two subsets
 * Z = the set of Integers (whole numbers)
 * the rest -- numbers that are fractions but not integers
 * finite decimals (decimals which stop after finite decimal places)
 * can be expressed as a fraction so they are rational
 * recurring decimals (infinite decimals with a repeating pattern)
 * can be expressed as a fraction so they are rational

... ... ... ... ... ...
 * Q ' = the set of Irrational Numbers
 * this includes all numbers that cannot be expressed as a fraction
 * so it is the complement of Q (the set of rational numbers)
 * when expressed as a decimal, there are infinite decimal places with no repeating pattern
 * Irrational numbers include
 * Special numbers like p
 * there are other special numbers you haven't met, like e, that are also irrational
 * non-recurring infinite decimals (decimals that go on for ever with no repeating pattern)


 * Z = the set of Integers (pronounced with a soft "g" as in giant)
 * Integers are the whole numbers
 * All integers are rational numbers so Z Ì Q
 * Integers can be divided up into:
 * Z – = the set of negative integers (not including zero)
 * Z + = the set of positive integers (not including zero)
 * Z + is also called the set of Natural numbers (N)
 * The Natural Numbers are all the counting numbers you learnt as a child
 * Because Z – and Z+ don't include zero, we have to list {0} separately

Note:
 * There are numbers that are not real
 * They cannot be located on a number line
 * They are called Imaginary Numbers (or Complex Numbers) and use //**i **//

math . \qquad i = \sqrt{-1} math

.
 * //**i **// can't possibly exist and so is completely imaginary and is not a Real number

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