Mathematics for Computing

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Mathematics for Computing
Lecture 7 Relations Dr Andrew
Purkiss-Trew Cancer Research UK e-mail
a.purkiss_at_mail.cryst.bbk.ac.uk
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Material to be covered today

• RelationsDefinition of a relationRepresentation
  of relationsEquivalence relationsPartial order
  relationsPartitioning

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Cartesian Products (from last time)

• A x B (x,y) x??A and y??B
• Example A 1,3,5, B 2,4A x B
  (1,2),(1,4),(3,2),(3,4),(5,2),(5,4)

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Relations

• If A and B are sets thenA binary relation R,
  from A to B assigns to each ordered pair (a,b) in
  A x B only (and only one) of the following
  statements
• i) a is related to b, written a R b
• ii) a is not related to b, written a R b

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Relations 2

• A relation R from A to B is a subset (R) of A x
  B
• R (a,b) a is related to b (a,b) a R
  bor any subset, R, of A x B, uniquely defines
  a relation R from A to B
• a R b whenever (a,b) ? R

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Relation of set on itself

• Relation on a set A is a subset of A x A.
• e.g. A 1,2,3,4, relation greater than
• (2,1),(3,1),(3,2),(4,1),(4,2),(4,3)

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Graphical Representation

• Directed Graph

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1
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Matrix Representation

• 1 2 3 41 F F F F2 T F F F3 T
  T F F4 T T T F

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Relation Definitions

• If R is a relation on a set A then
• R is reflexive if x R x for all x ? A
• R is irreflexive if there are no elements of A
  for which x R x
• R is symmetric if x R y implies y R x, for all x,
  y ? A
• R is antisymmetric if x R y and y R x imply x
  y, for all x, y ? A
• R is transitive if x R y and y R z imply x R z,
  for all x, y, z ? A

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Equivalence Relation

• An equivalence relation is one that is
• reflexive
• symmetric and
• transitive

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Partial Order Relation

• An partial order relation is one that is
• reflexive
• antisymmetric and
• transitive

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Partitioning and equivalence

• If A is a set, then a partition of A is a set of
  subsets such that every element in A is an
  element of exactly one of the subsets.
• If A is a set and R is an equivalence relation on
  A. For each element x of A, let E(x) be the
  subset of A defined E(x)y?A y R x. The set
  of all E(x) is a partition of A. The subsets
  E(x) are the equivalence classes of R