Relations

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Relations
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Ordered Pairs

• An ordered pair is like a set of 2 elements
  except that order matters (a, b) ? (b, a).
• Let X Y be sets.
• The Cartesian product or cross product denoted
• X ? Y (x, y) x ? X, y ? Y.
• A binary relation on X to Y is a subset of X ? Y.

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Ordered Pairs ...

• Example C (x, y) x,y ?R, x2 y2 lt 1.
• Is C a relation?
• If so, on what 2 sets?
• Describe in English the relationship between x
  and y.
• Example G (1, 17), (56, 100003) is a
  relation on N ? N.
• The mathematical definition of relation does not
  require that the ordered pairs have any
  meaningful relationship.

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Binary Relations

• Example Let T a, b, c, d, e, f be the set of
  positions on a circuit board where chips are
  placed.
• Draw a line between any 2 positions whose chips
  are connected by a wire.
• The set C of ordered pairs of chips that are
  connected by a wire is a relation on T ? T

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Circuit board example ...

• (a, b), (b, a),
• (b, c), (c, b),
• (b, d), (d, b),
• (c, d), (d, c),
• (d, f), (f, d)
• In this relation, if (x, y) ? C then (y, x) ? C.
• Such a relation is called symmetric.
• A relation that is not symmetric is called
  asymmetric.

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Circuit board example ...

• If, in the circuit board example, the chips were
  diodes and current runs in only 1 direction, then
  another relation could be the set of ordered
  pairs of chips (x, y) where current runs from x
  to y.
• Would this relation be symmetric?

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Antisymmetric

• A binary relation R ? A?A is antisymmetric when
  aRb ? bRa ? a b.
• For numbers, which are antisymmetric?
• a ? b
• a lt b
• For sets, which are antisymmetric?
• A ? B
• A ? B
• Is there a difference between asymmetry and
  antisymmetry?

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Binary relations represented as graphs

• Turning things around
• Given a binary relation B ? X ? Y, represent B as
  a directed graph G (X?Y, B)
• the nodes of the G are the elements of X Y
• there is an arc from x to y if (x, y) ? B.

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N-ary Relations

• Cartesian products are extended to n sets
• Y X1 ? X2 ? ? Xn.
• A subset of Y is an n-ary relation.
• Such relations play a central role in relational
  database theory.
• n-ary relations can be represented as hypergraphs
  (not a handy visualization).