SETS

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SETS

• A set B is a collection of objects such that for
  every object X in the universe the statement
• X is a member of B
• Is a proposition.

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A quick review of basic notation and set
operations.

• A 1, 2, ab, ba, 3, moshe, table,
• 1,2, ab, ba, moshe table are elements. They are
  members of the set A or belong to A.
• Notation ab ? A a ? A
• 3. V a, i, o, u, e Set of Vowels
• O 1,3,5,7,9 Odd numbers lt 10.
• 4. A1 2, 5, 8, 11, , 101
• A2 1, 2, 3, 5, 8, 13,
• A3 2, 5, 10, 17, 26, , 101

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Basic notation.

• Set Builder B x P(x)
• B1 x x n2 1, 1 ? n ? 10 (B1 A3)
• B2 p p prime, p n! 1, n ????
• Special sets
• N (non-negative integers, natural numbers)
• Q (rational numbers)
• Z (integers)
• Z (positive integers)
• R (real numbers)
• ? (the empty set)

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Relations among sets

• A B if x ? A ? x ? B
• Subsets A ? B A ? B A ? B A ? B
• For every set B ? ? B
• A set may have other sets as members
• A ?, a, b, a,b. Note A has 4
  elements. ? ? A and also ? ? A, a ? A, a ? A,
  a ? A.

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Set operations

• Union A ? B x x ? A ? x ? B (logic or)
• Intersection A ? B x x ? A ? x ? B (
  and)
• Set difference A \ B x x ? A ? x ? B
• Complement of A A a a ? A or if U is the
  universe then A U \ A (not).
• Example If U a,b,c,,z and A i,o,e,u,a
  then A n n is not a vowel.
• Symmetric difference A ? B (A \ B) ? (B \
  A) (xor)

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• The characteristic vector of a set
  (representing sets in memory)
• Let U 1,2,, 15. Let A 3,5,11,13 the
  characteristic vector of A is the binary string
  00101 00000 10100.
• The characteristic vector 10010 01101 10001
  represents the set 1,4,7,8,10,11,15.
• 00000 00000 00000 represents ?.
• Note with this representation the union of two
  sets is the OR bit operation and the intersection
  is the AND.

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A simple application.

• Problem find the smallest integer n that
  satisfies the following 3 conditions
  simultaneously
• (n mod 7 5), (n mod 11 7), (n mod 17
  9)
• Knowing the language Math can help us look for
  information and use various systems to solve this
  problem. The following exolains how to use SAGE's
  set operations to solve problems.

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We can create three sets 1. A k k 7n
5, k lt 4000 2. B k k 11n 7, k
lt 4000 3. C k k 17n 9, k lt
4000 We can then ask SAGE to find the
intersection of the three sets. The smallest
integer in the intersection (provided there is
one) will be our solution. Answer 502, 1811,
3120, ...
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Venn Diagrams

• Venn Diagrams a useful tool for representing
  information. For instance, the various sets that
  can be formed by the basic set operations can be
  viewed by a Venn Diagram.

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A
B
C
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• Proving set equalities
• Either x ? A ? x ? B or A ? B ? B ? A.
• Example De Morgans law A ? B A ? B
• Proof Let x? A ? B.
• Then x ? A ? B.
• Or x ? A and x ? B
• Or x ?? A and x ? B
• Or x ?? A??? B
• Conversely, start from the bottom and go up.
• QED
• Assume M 1,2,5,9 then
• A1 ? A2 ? A5 ? A9

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• Notation
• A1 ? A2 ? ? An x x ? Ai i 1, 2, , n.
• A1 ? A2 ? ? An x ? ( i, 1??i ?? n) x ? Ai.
• Use formula to insert intersection.
• Assume M 1,2,5,9 then
• A1 ? A2 ? A5 ? A9

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The Power Set

• Definition The Power set of the set A is
• P(A) B B ? A.
• ??has 0 elements. P(?) has one element P(?)
  ?
• A a P(A) ?, a P(?) ?, ?

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The cartesian product

• Cartesian product
• A x B (a,b) a ? A ? b ? B
• Can be defined using sets only
• A x B a, a,b a ? A ? b ? B
• Note (a,b) ? (b,a) if a ? b.
• Cartesian product of n sets A1x A2 x x An
  (a1, a2,, an) ai ? Ai, i 1,,n

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Relations

• Definition 1 A relation R, (binary relation)
  between two sets A and B is a subset of
• A x B (mathematically speaking R ? A x B).
• Definition 2 A relation R on a set A is a subset
  of A x A.

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Relations

• There are two common ways to describe relations
  on a set or between two sets
• List all pairs belonging to the relation.
• Use set builders to describe the pairs.
• Example 1 R0 (4,3), (9,2), (3,6), (7,5) is
  a relation on N. It is also a relation on A x B
  where A 4,9,3,7 and B 3,2,6,5

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More examples

• Example 2 R2 (n,k) n ? N and n k is a
  prime number.
• Example 3 R3 (n,k) n,k ? N and n k is
  a multiple of 19.
• Example 4 R3 (w,m) w is a woman, m is a
  man, w dates m

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Classification of relations

• These definitions apply to relations on A.
• Definition 3 A relation R on A is reflexive
• if (a,a) ? R ?a ? A.
• Definition 4 A relation R on A is symmetric
• if (a,b) ? R then (b,a) ? R.
• R is antisymmetric
• if (a,b) ? R and (b,a) ? R only if a b.
• Definition 5 A relation R on a set A is
  transitive
• if (a,b) ? R ? (b,c) ? R then (a,c) ? R.

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The transitive closure

• Observation If R1 and R2 are transitive
  relations on a set A then so is R1 ? R2.
• Proof Obvious.
• Definition 6 The transitive closure of a
  relation R on a set A is the smallest
  transitive relation R on A such that R ? R.

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I think I solved it!