Definitions for Binary Relations over A x A
A majority of the binary relations we will be dealing with are a
subset of the Cartesian product of a particular set with itself.
If we have R
⊆
A x A, then we have the following definitions:
1)
R is reflexive if
2200
a
∈
A, (a,a)
∈
R.
2)
R is irreflexive if if
2200
a
∈
A, (a,a)
∉
R.
3)
R is symmetric if
2200
a
∈
A, aRb
⇒
bRa
4)
R is antisymmetric if aRb
∧
bRa
⇒
a=b.
5)
R is transitive if
aRb
∧
bRc
⇒
aRc.
Consider the following relation R defined over {a , b, c}:
R = { (a,b), (a,c), (b,a), (b,c), (c,c) }
R is not reflexive since (b,b)
∉
R
R is not irreflexive since (c,c)
∈
R
R is not symmetric since we have (a,c)
∈
R, but (c,a)
∉
R.
R is not antisymmetric since (a,b)
∈
R and (b,a)
∈
R.
R is not transitive since (b,a)
∈
R, (a,b)
∈
R, but (b,b)
∉
R.
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View Full DocumentNow, I will show you some examples of more meaningful
relations that actual have some of these properties.
Consider a relation R over the set {beans, rice, chicken} that is
defined as foods that go well together. The relation could be:
R = { (beans, beans), (rice, rice), (chicken, chicken), (beans,
rice), (rice, chicken), (rice, beans), (chicken, rice) }
This relation is reflexive since for each element a, (a,a)
∈
R.
Essentially, we can mix anything with itself and it’ll still be
edible.
This relation is also symmetric. The reason for this is that if we
can mix one food first with a second food, then we can ALSO
mix the second food with the first. Symbolically, for each pair
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 Spring '09
 Equivalence relation, Transitive relation, relation, aRb

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