Representing Relations

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Representing Relations

• Rosen 6.3

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Using Matrices

• For finite sets we can use zero-one matrices.
  Elements of each set A and B must be listed in
  some particular (but arbitrary) order. When AB
  we use the same ordering for A and B.
• mij 1 if (ai,bj) ??R
• 0 if (ai,bj) ?R

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Example Zero-One Matrix
b1 b2 b3
a1 a2 a3
R (a1,b1), (a1,b2), (a2,b2), (a3,b2), (a3,b3)
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Matrix of a relation on a set, A

• Can be used to determine whether the relations
  has certain properties.
• Recall that R on A is reflexive if (a,a) ?R for
  every element a? A.

Reflexive Not Reflexive
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A relation R on a set A

• is called Symmetric if (b,a) ?R whenever (a,b) ?R
  for a,b ?A. MR (MR)t
• Is antisymmetric (a,b) ?R and (b,a) ?R only if
  ab for a,b ?A is antisymmetric.
• If mij 1, i?j, mji 0

Symmetric Antisymmetric Neither
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Examples
Reflexive Symmetric
Reflexive Antisymmetric
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Let R1, R2 be relations on A

• A 1,2,3
• R1 (1,1), (1,3), (2,1), (3,3)
• R2 (1,1), (1,2), (1,3), (2,2), (2,3), (3,1)

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R1?R2, R1?R2
MR1?R2 MR1 ? MR2, MR1?R2 MR1 ? MR2
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What is R1 ? R2?

• The composite of R1 and R2 is the relation
  consisting of ordered pairs (a,c) where a ? A, c
  ? A, and for which there exists an element b ? A
  such that (a,b) ? R1 and (b,c) ? R2.
• R1 ? R2 (1,1), (1,2), (1,3), (3,1), (2,1),
  (2,2), (2,3)

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Boolean Product

• Let A aij be an m by k zero-one matrix and B
  bij be a k by n zero-one matrix. Then the
  Boolean Product of A and B denoted by A B is
  the m by n matrix with i,j entry cij where
• cij (ai1?b1j) ? (ai2 ? b2j) ?... ? (aik ? bkj).

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What is R1 ? R2?

• R1 ? R2 (1,1), (1,2), (1,3), (3,1), (2,1),
  (2,2), (2,3)
• MR1?R2 MR1 MR2

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Directed Graphs (Digraph)

• A directed graph consists of a set V of vertices
  together with a set E of ordered pairs of
  elements of V called edges.
• (a,b), a is initial vertex, b is the terminal
  vertex

Reflexive (Loops) Symmetric (Edges both ways)
b
a
c
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Relation R on a set A
R (a,b), (b,b), (b,c), (c,a),
(c,c) Transitive
b
a
c
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Relation R on a set A
R (a,a), (a,c), (b,b), (b,a), (b,c),
(c,c) Reflexive Antisymmetric
b
a
c