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Set: A well-defined collection of distinct objects.

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d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9, pw, sw, c, p

Curly braces (brackets)

{ }

Elements

Set NotationRoster Method

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{ d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9, pw, sw, c, p }Universal Set - U

The set consisting of all of the elements to be considered.

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A = { d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9, pw, sw, c, p }

B = { d, 3w, 5w, 7w }

B is a subset of AB A

A

B

One set (B) is a subset of another (A) if EVERY element of set B is contained in set A and is written B A

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A = { d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9}

B = { 4, 5, 6, 7, 8, 9, pw, sw, c, p }

{d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9 pw, sw, c, p }A B =

Union of Sets

Union: The set of elements that belong to either set A or set B or to both and is written A B

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A = { d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9}

B = { 4, 5, 6, 7, 8, 9, pw, sw, c, p }

{ 4, 5, 6, 7, 8, 9 }A B =

Intersection of Sets

Intersection : The set of elements that belong to BOTH set A AND set B is denoted A B.

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Null Set(empty Set)

{ }

Not zero!!!!

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Venn Diagrams

Universal Set

BA

A Bsubset

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Venn Diagrams

BA BUunion

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Venn Diagrams

BA B

Uintersection

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Roster Method & Set-Builder Notation

A = { d, 3w, 5w, 7w, 4, 5, 6, 7, 8, 9, pw, sw, c, p }

roster method

A = { x | x is a club in the golf bag }

set-builder notation

Read “such that”

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Roster Method & Set-Builder Notation

A = { 6, 7, 8, 9, … }

roster method

A = { x | x is an integer > 5 }

set-builder notation

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Natural Numbers

Natural or counting numbers.

{1, 2, 3, …}Closed under addition.

Example: 5 + 2 = 7 which is a natural number!

Natural Numbers

Natural or counting numbers.

{1, 2, 3, …}Closed under subtraction?

Example: 2 – 5 = ? a natural number?

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Integers

{0, ±1, ± 2, ± 3, …}

{0, ±1, ± 2, ± 3, …}{1, 2, 3,…}

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Integers

{0, ±1, ± 2, ± 3, …}

Closed under addition & subtraction!Closed under multiplication?Closed under division?

, ,

Rational Numbers

=

{ |m, n , n≠0 }

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Rational Numbers

Closed under +, -, x, ÷

Denominator NEVER zero!!

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Decimal expansions:

Terminating ⅖ = .4Or

Non-terminating repeating ⅙ = .161616…

Rational Numbers Expressed as Decimals

3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798…

Irrational NumbersAn irrational number is a number that cannot be expressed as a fraction.

Decimal expansions:

Do NOT terminate.AND

Do NOT repeat.

Pi = π

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à is irrational !

1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727350138…

≈ 1.41

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Universe of Real NumbersReal Numbers

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Real Number Line

1 2 3

-1 0 π

=U

-

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Real Number Line-

Points Real Numbers1 to 1

-Is complete – (no holes)-Is ordered