Function vs. Relation Composition
I looked over the book and my notes and found a discrepancy
between the two. In particular, I had told you that when you
compose a 2 relations, for example R
⊂
A
×
B, S
⊂
B
×
C,
that this composition would be written, just as if R and S were
functions:
S
°
R = { (a,c)  a
∈
A
∧
c
∈
C
∧
(
5
b  (a,b)
∈
R
∧
(b,c)
∈
S) }
But, composition with relations, is written the other way
around, so that it is “intuitive” given the graph of the relation.
Thus, the definition we have for relation composition with the
2 relations above is as follows:
R
°
S = { (a,c)  a
∈
A
∧
c
∈
C
∧
(
5
b  (a,b)
∈
R
∧
(b,c)
∈
S) }
BUT, if we were composing two functions f : A
→
B and g : B
→
C, the definition remains as I showed you:
g
f = {(a, c)  a
∈
A
∧
c
∈
C
∧
(
5
b
∈
B  (a, b)
∈
f
∧
(b, c)
∈
g)}.
With that in mind, and the idea that I should separate my
presentation of relations and functions, I have reorganized the
notes from my past three lectures.
I apologize for the inconvenience, but I feel that this
organization will clarify the material presented over the past
three lectures.
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Last lecture, I asked you all to find a counterexample to the
following (note that I have written this correctly, for relations
R, S and T defined over the sets A, B and C as described in the
last lecture):
(R
S)
∩
(R
T)
⊂
R
(S
∩
T),
where R: A
→
B and S,T: B
→
C.
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 Fall '09
 Binary relation, Transitive relation, a,b, Transitive closure

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