Problems involving relation composition
Prove or disprove: If a relation R, defined as a subset of A x A, is
symmetric, then show that R
°
R is symmetric as well.
Since R is symmetric, we know that if (a,b)
∈
R, then (b,a)
∈
R.
We must prove under this assumption that if (a,b)
∈
R
°
R, then
(b,a)
∈
R
°
R.
If (a,b)
∈
R
°
R, by the definition of function composition, we
must have an element c
∈
A such that (a,c)
∈
R and (c,b)
∈
R.
But we know R is symmetric. Thus, we can deduce that
(c,a)
∈
R and (b,c)
∈
R.
But, if this is the case, by the definition of function
composition, we must have (b,a)
∈
R
°
R, because (b,c)
∈
R and
(c,a)
∈
R. This is exactly what we needed to prove. Thus, R
°
R is
symmetric.
Prove: R is transitive
⇒
R
R
⊂
R.
We must show that if (a,b)
∈
R
°
R, then (a,b)
∈
R.
First of all, if (a,b)
∈
R
°
R, then by the definition of function
composition, there exists an element c of A such that (a,c)
∈
R
and (c,b)
∈
R.
But, we also know that R is transitive. Since we have (a,c)