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show that R3 o (R2 o R1) = (R3 0 R2) o R1. TI H

show that R3 o (R2 o R1) = (R3 0 R2) o R1. TI H

Advanced Engineering Mathematics
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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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 0 (R2 o R1) (R3 o 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 Ax 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) EC x 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) C (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 0 (R2 o R1).)
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 0 (R2 o R1) (R3 o 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 Ax 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) EC x 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) C (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 0 (R2 o R1).)
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