Math Camp 2005 Instructor: Udi Sommer

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Math Camp 2005Instructor Udi Sommer

• First Class (Tuesday, August 16, 3-6p)
• introduction
• basic rules of algebra
• numbers
• properties of numbers
• sets
• notation

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1. Basic Rules of Algebra

• Cartesian coordinate system
• Ratios, Fractions
• a / b a ? (1 / b)
• a / b c / d ((a ? d) (b ? c)) / (b ? d)
• a / b c / d a ? c / b ? d
• a / b c / d a ? d / b ? c
• Signs
• a ? (-b) - (a ? b)
• (-a) ? (-b) a ? b
• (-a) a

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1. Basic Rules of Algebra

• Exponents
• Logarithms
• Commutative, associative and distributive rules
• x y y x --- Commutative (addition and
  multiplication)
• (x y) z x (y z) --- Associative
  (addition and multiplication)
• x (y z) x y x z
• Useful Formulas
• (x y)² x² 2xy y²
• (x y) ² x² - 2xy y²
• (x y)(x y) x² - y²

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Numbers

• Natural / Counting numbers N 1, 2, 3, 4 -
  nonnegative integers
• Integers I -2, -20, 5 - negative and
  nonnegative integers
• Rational (Ratio of integers) Q 5, ½, ¾ - numbers
  that can be expressed in a ratio or faction
  numerator denominator can be expressed in
  decimal notation - ¾ .75, or others which do
  not terminate in decimal notations but rather
  repeat - ? .33333
• Real numbers ? 2.718, v3 - these include real
  numbers as well as irrational numbers (those that
  cannot be written in the form of a ratio or
  faction), other e.g.s p 3.1415926 , e
  (Eulers number) 2.7182 Number Line

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3. Properties of Numbers

• Ordinal Properties of Numbers refer to order /
  to ranking
• Inequalities greater / less than 3gt1,
  -45lt508Laws of inequalities 1. if a lt b then a
  c lt b c2. if a lt b and c gt 0, then a?c lt
  b?c3. if a lt b and c lt 0, then a?c gt b?c4. if a
  gt 0 then 1/a gt 05. if a, b ? ?, and if a lt b
  then 1/a gt 1/b6. Transitivity --- if agtb and
  bgtc, then agtc

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3. Properties of Numbers

• Distance along a lineabsolute value ?a-b?
  the distance along a line between a and b
• Euclidean distance in a plane v(a1 b1)²
  (a2 b2)²
• Means / AveragesArithmetic mean (a b) /
  2Geometric mean - v(a b)

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4. Sets

• A set
• finite sets, and infinite sets
• or Ø is the empty set
• Members or elements
• Universe U
• Equation
• Subset
• Proper Subset
• An event
• A complement
• Intersection
• Union

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Venn Diagrams
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Set Rules and Theorems

• Closure for each pair of sets A and B, there
  exists a unique A ? B and a unique A ? B.
• Commutativity -Union and intersection are
  commutative -
• A ? B B ? A A ? B B ? A
• Associativity - Union and intersection are
  commutative
• (A ? B) ? C A ? (B ? C)
• (A ? B) ? C A ? (B ? C)
• Distributivity - A ? (B ? C) (A ? B) ? (A ?
  C)
• A ? (B ? C) (A ? B) ? (A ? C)
• Identity A ? U A for each set A ? U, and
  there exists a unique set ? such that A ? ? A
  for each set A
• Complementation for each set A ? U there exists
  a unique set A such that A ? A ? and A ? A
  U

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Set Rules and Theorems

• Idempotency -
• A ? A A ? A A
• A ? U A ? ? A
• A ? U U
• A ? ? ?
• DeMorgans Laws
• (A \ B) ? (A \ C) A \ (B ? C)
• (A \ B) ? (A \ C) A \ (B ? C)
• (A ? B) A ? B
• (A ? B) A ? B
• A ? A U
• A ? A ?
• Absorption Laws
• A ? A ? B A
• A ? (A ? B) A
• Miscellaneous
• (A ? B) ? (A \ B)A
• (A \ B) ? B A ? B
• A ? A ? B

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5. Notations

• ? - summation
• ? - product
• ? - there exists
• ? - therefore
• ? - for all