Sets and their operation

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Section 3.1

• Sets and their operation

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Definitions

• A set S is collection of objects. These objects
  are said to be members or elements of the set,
  and the shorthand for writing x is an element of
  S is x ? S.
• The easiest way to describe a set is by simply
  listing its elements (the roster method). For
  example, the collection of odd one-digit numbers
  could be written 1, 3, 5, 7, 9. Note that this
  is the same as the set 9, 7, 5, 3, 1 since the
  order elements are listed does not matter in a
  set.

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Examples

• The elements of a set do not have to be numbers
  as the following examples show
• Doug, Amy, John, Jessica
• TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF
• A,B, A,C, B,C

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Common sets of numbers

• Page 182
• N set of natural numbers 0, 1, 2,
• Z set of integers , -2, -1, 0, 1, 2,
• Q set of rational numbers
• R set of real numbers

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Definitions

• If A and B are sets, then the notation A ? B
  (read A is a subset of B) means that every
  element of set A is also an element of set B.
• Practice. Which is true?
• 1, 2, 3, 4 ? 2, 3, 4
• Z ? Q
• Z ? N
• ? a, b, c
• 3, 5, 7 ? 2, 3, 5, 7, 11
• a, b ? a, b, a, c, b, c
• a ? a, b, a, c, a, b, c

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Set notation

• Large sets cannot be listed in this way so we
  need the more compact set-builder notation.
  This comes in two types exemplified by the
  following
• (Property) n ? Z n is divisible by 4
• (Form) 4k k ? Z

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Practice with property description

• List five members of each of the following sets
• n ? N n is an even perfect square
• x ? Z x 1 is divisible by 3
• r ? Q r2 lt 2
• x ? R sin(x) 0

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Practice with form description

• List five members of each of the following sets
• 3n2 n ? Z
• 4k 1 k ? N
• 3 2r r ? Q and 0 ? r ? 5

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Definitions of set operations

• Let A and B be sets with elements from a
  specified universal set U.
• A ? B (read A intersect B) is the set of
  elements in both sets A and B.
• A ? B (read A union B) is the set of elements
  in either set A or B.
• A B (read A minus B) is the set of elements
  in set A which are not in B.
• A (read the complement of A) is the set of
  elements in the universe U which are not in A.

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Practice with set operations

• Let A 1, 3, 5, 7, 9, B 2, 4, 6, 8, 10,
• C 2, 3, 5, 7, D 6, 7, 8, 9, 10
• be sets with elements from the universal set U
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10
• Find each of the following

• A ? C
• B ? D
• B D
• B
• (A ? B) C

• (A ? C) ? B
• B ? C
• (B ? C)
• (C ? D) A
• B ? D

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Venn diagrams
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Inclusion-Exclusion Principle

• The notation n(A) means the number of elements
  of A. For example, if A 2, 3, 6, 8, 9, then
  n(A) 5.
• Principle of Inclusion/Exclusion for two sets A
  and B
• n(A ? B) n(A) n(B) n(A ? B)

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Inclusion-Exclusion Principle

• Example. A 2, 4, 6, 8, , 96, 98, 100 and
  B 5, 10, 15, 20, , 90, 95, 100
• n(A ? B) n(A) n(B) n(A ? B)
• 50 20 10
• 60

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Inclusion-Exclusion Principle

• Principle of Inclusion/Exclusion for three sets
  A, B, and C
• n(A ? B ? C) n(A) n(B) n(C)
• n(A ? B) n(A ? C) n(B ? C)
• n(A ? B ? C)