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
2200
a
∈
A, (a,a)
∉
R.
3)
R is symmetric if
2200
a
∈
A, aRb
⇒
bRa
4)
R is anti-symmetric 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 anti-symmetric since (a,b)
∈
R and (b,a)
∈
R.
R is not transitive since (b,a)
∈
R, (a,b)
∈
R, but (b,b)
∉
R.