at R3 0 (R2 0 R1) = (R3 0 R2) o R1. Thi
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1.
Suppose that we have the binary relations R1 CAx B, R2 C B×C,
and R3 C C x D. Then the relations R3 0 (R2 0 R1) and (R3 0 R2) o R1 are both relations
from A to D. In this exercise, you want to show that they are in fact the same relation, i.e.
R3 0 (R2 o R1) = (R3 0 R2) o R1.
Recall that
• R2 o Rị is the set of all ordered pairs (a, c) E A× C such that, for each of them, there
exists a b e B such that (a, b) e Rị and (b, c) e R2,
R3 0 (R20 R1) is the set of all ordered pairs (a, d) E Ax D such that, for each of them,
there exists a c E C such that (a, c) E R2 0 R1 and (c, d) E R3,
• R3 o R2 is the set of all ordered pairs (b, d) E B × D such that, for each of them, there
exists a c E C such that (b, c) E B × C and (c, d) e C × D,
(R30 R2) o R1 is the set of all ordered pairs (a, d) E Ax D such that, for each of them,
there exists a b e B such that (a, b) E A × B and (b, d) E R2 × R3.
Use these four summary points to show that R3 o (R2 o R1)
(R3 o R2) o R1. This can be
shown by showing two things:
first, show that each point in R3 o (R2 o R1) is also in (R3 0 R2) o R1, and
second, the converse, where each point in (R3 o R2) o R1 is also in R3 o (R2 o R1), is also
true.
(This is often the way to show that two sets are equal. The first part shows that R3 0 (R2 o
R1) (R3 0 R2) o R1 and the second part shows that (R3 0 R2) o R1 C R3 0 (R2 o R1),
whence (R3 o R2) o R1
R3 o (R2 0 R1).)
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