Let R and S be relations on {a, b, c, d}, whereR = {(a, b), (a, d), (b, c), (c, c), (d, a)}andS = {(a, c), (b, d), (d, a)}.Then,| R o S | + | (S o R)^(-1) | =
Question
Let R and S be relations on {a, b, c, d}, where
R = {(a, b), (a, d), (b, c), (c, c), (d, a)}
and
S = {(a, c), (b, d), (d, a)}.
Then,
| R o S | + | (S o R)^(-1) | =
Solution
First, let's understand the operations involved in the question.
-
RoS (R composition S) is a relation from S to R such that for any elements x, y, z, if (x, y) is in S and (y, z) is in R, then (x, z) is in RoS.
-
(SoR)^-1 is the inverse of the relation SoR. If (x, y) is in SoR, then (y, x) is in (SoR)^-1.
Now, let's calculate RoS and (SoR)^-1.
RoS = {(x, z) | there exists a y such that (x, y) is in S and (y, z) is in R} = {(a, b), (a, d), (b, c), (d, a)}
|RoS| = 4
SoR = {(x, z) | there exists a y such that (x, y) is in R and (y, z) is in S} = {(b, d), (d, c), (a, a)}
(SoR)^-1 = {(d, b), (c, d), (a, a)}
|(SoR)^-1| = 3
Therefore, |RoS| + |(SoR)^-1| = 4 + 3 = 7.
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