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Friday, 5 April 2013

Binary Relations and its Properties

Symmetric
a,baRbbRa
where R is a relation. For example,
  • = relation on R
  • is equal to
  • is married to 
  • {(1,2), (2,1)}
Antisymmetric
a,b:XaRbbRaa=b
For example,
  • {(1, 2), (2, 3), (1, 1)}
Asymmetric
a,b:XaRb¬(bRa)
For example,
  • {(1,2), (3,4)}
Reflexive
x:XxRx
For example,
  • =
Irreflexive (strict)
x:X¬(xRx)
For example,
  • >, <

Coreflexive (a subset of Identity Relation)
x,y:XxRyx=y)
For example,
  •  equal to and odd, such as {(1,1), (3,3)}
Identity
{(x,x)|xX}

Transitive
x,y,z:XxRyyRzxRz
For example,
  • <, =,



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