1. Write out the addition and multiplication tables for Z6 (where by addition and multiplicationwe mean +6 and 6).2. Let f : Z? –→ Z² be given by f(m,n) = (m-n,n). The composite functions fk, for k e Z+, aredefined as fi(m, n) = f(m,n), and fk+1(m,n) = f(fr(m, n)), for k e Z*. Give a formal proofby induction that fr(m,n) = (m – kn, n) , for all k e Z+.-3. Use induction to show that for all positive integers n(a) 13 + 23 + 33 + ... + n³ = (n(n + 1)/2)².(b) 1·1! + 2 - 2! + ... + n· n! = (n + 1)! – 1(c) if n > 6, then 3" < n!

Addition in modulo 6 is defined as a +6 b=a+bif a+b<6a+b-6if a+b≥6 Multiplication in modulo 6 is…

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1. Write out the addition and multiplication tables for Z6 (where by addition and multiplication we mean +6 and 6).

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 61EQ

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Question

1. Write out the addition and multiplication tables for Z6 (where by addition and multiplication
we mean +6 and 6).
2. Let f : Z? –→ Z² be given by f(m,n) = (m-n,n). The composite functions fk, for k e Z+, are
defined as fi(m, n) = f(m,n), and fk+1(m,n) = f(fr(m, n)), for k e Z*. Give a formal proof
by induction that fr(m,n) = (m – kn, n) , for all k e Z+.
-
3. Use induction to show that for all positive integers n
(a) 13 + 23 + 33 + ... + n³ = (n(n + 1)/2)².
(b) 1·1! + 2 - 2! + ... + n· n! = (n + 1)! – 1
(c) if n > 6, then 3" < n!

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