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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