1. Suppose that we have the binary relations R1 C A× B, R2 C Bx C, and R3 C C x D. Then the relations R3 o (R2 o R1) and (R3 o 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 o (R2 o R1) = (R3 0 R2) o R1. Recall that • R2 o R1 is the set of all ordered pairs (a, c) E Ax C such that, for each of them, there exists a b e B such that (a, b) E R1 and (b, c) e R2, • R30 (R20 R1) is the set of all ordered pairs (a, d) € 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) EB x 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 x D, (R3o 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 x B and (b, d) E R2 x R3. Use these four summary points to show that R3 o (R2 0 R1) = (R3 0 R2) o R1. This can be %3D shown by showing two things: first, show that each point in R3 0 (R2 o R1) is also in (R3 o 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 o (R2 o R1) C (R3 0 R2) o R1 and the second part shows that (R3 o R2) o R1 C R3 o (R2 o R1), whence (R3 o R2) ● R1 = R3 0 (R2 0 R1).) 2. Using the result above, show using induction that, if R is a relation on A, then R" o R = Ro R" for each n E N. (Hence, we may define the Nth iterate of R either way.)
1. Suppose that we have the binary relations R1 C A× B, R2 C Bx C, and R3 C C x D. Then the relations R3 o (R2 o R1) and (R3 o 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 o (R2 o R1) = (R3 0 R2) o R1. Recall that • R2 o R1 is the set of all ordered pairs (a, c) E Ax C such that, for each of them, there exists a b e B such that (a, b) E R1 and (b, c) e R2, • R30 (R20 R1) is the set of all ordered pairs (a, d) € 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) EB x 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 x D, (R3o 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 x B and (b, d) E R2 x R3. Use these four summary points to show that R3 o (R2 0 R1) = (R3 0 R2) o R1. This can be %3D shown by showing two things: first, show that each point in R3 0 (R2 o R1) is also in (R3 o 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 o (R2 o R1) C (R3 0 R2) o R1 and the second part shows that (R3 o R2) o R1 C R3 o (R2 o R1), whence (R3 o R2) ● R1 = R3 0 (R2 0 R1).) 2. Using the result above, show using induction that, if R is a relation on A, then R" o R = Ro R" for each n E N. (Hence, we may define the Nth iterate of R either way.)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 32EQ
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Transcribed Image Text:1.
Suppose that we have the binary relations R1 C Ax B, R2 C B×C,
and R3 C C x D. Then the relations R3 o (R2 o R1) and (R3 o 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 o (R2 o R1) = (R3 0 R2) o R1.
Recall that
• R2 o R1 is the set of all ordered pairs (a, c) E Ax C such that, for each of them, there
exists a b e B such that (a, b) E Rị and (b, c) E R2,
• R30 (R20 R1) is the set of all ordered pairs (a, d) e A× 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 x 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,
(R3o R2) o R1 is the set of all ordered pairs (a, d) E A× 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 0 R2) o R1. This can be
%3D
shown by showing two things:
first, show that each point in R3 0 (R2 o R1) is also in (R3 o 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 o (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) ● R1
R3 o (R2 0 R1).)
2.
Using the result above, show using induction that, if R is a relation on
A, then R" o R = Ro R" for each n E N. (Hence, we may define the Nth iterate of R either
way.)
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