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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 (R20 R1) and (R3 0 R2) ● 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 • R2o Rị is the set of all ordered pairs (a, c) € A × C such that, for each of them, there exists a b e B such that (a, b) E Rị and (b, c) € R2, • R3o (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) € R2 0 R1 and (c, d) E R3, • R3o 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 Bx 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 0 (R2 0 R1) = (R3 0 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 Rị is also in R3 0 (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) C (R3 0 R2) o R1 and the second part shows that (R3 o R2) o R1 C R3 0 (R2 o R1), whence (R3 o R2) o R1 = R3 0 (R2 o R1).) %3D

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 (R20 R1) and (R3 0 R2) ● 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 • R2o Rị is the set of all ordered pairs (a, c) € A × C such that, for each of them, there exists a b e B such that (a, b) E Rị and (b, c) € R2, • R3o (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) € R2 0 R1 and (c, d) E R3, • R3o 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 Bx 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 0 (R2 0 R1) = (R3 0 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 Rị is also in R3 0 (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) C (R3 0 R2) o R1 and the second part shows that (R3 o R2) o R1 C R3 0 (R2 o R1), whence (R3 o R2) o R1 = R3 0 (R2 o R1).) %3D

Linear Algebra: A Modern Introduction
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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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 0 (R2 0 R1) and (R3 0 R2) ● 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, • 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 o 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 0 R1) = (R3 0 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) C (R3 0 R2) ● R1 and the second part shows that (R3 o R2) o R1 C R3 0 (R2 o R1), whence (R3 o R2) o R1 = R3 0 (R2 o 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.)
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 0 (R2 0 R1) and (R3 0 R2) ● 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, • 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 o 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 0 R1) = (R3 0 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) C (R3 0 R2) ● R1 and the second part shows that (R3 o R2) o R1 C R3 0 (R2 o R1), whence (R3 o R2) o R1 = R3 0 (R2 o 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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