Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 5 - Number Theory and the Real Number System - 5.5 Real Numbers and Their Properties; Clock Addition - Exercise Set 5.5 - Page 309: 50


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Work Step by Step

(a) The set\[\left\{ 0,1,2,3,4,5,6 \right\}\]is closed under the operation of clock addition, because the entries of the table are all the elements of the set. Hence, the set \[\left\{ 0,1,2,3,4,5,6 \right\}\]is closed under the operation of clock addition. (b) Consider the expression of LHS. \[\begin{align} & \left( 3\oplus 5 \right)\oplus 6=1\oplus 6 \\ & =0 \end{align}\] Now, find RHS as: \[\begin{align} & 3\oplus \left( 5\oplus 6 \right)=3\oplus 4 \\ & =0 \end{align}\] Hence, \[\left( 3\oplus 5 \right)\oplus 6=3\oplus \left( 5\oplus 6 \right)\]. Associative property has been verified. (c) Identity element is that element, which does not change anything when clock addition is used. Now, as we can see that, \[0\oplus 0=0,\ 0\oplus 1=1,\ 0\oplus 2=2,\ 0\oplus 3=3,\ 0\oplus 4=4,\ 0\oplus 5=5,\ 0\oplus 6=6\]and, \[0\oplus 0=0,\ 1\oplus 0=1,\ 2\oplus 0=2,\ 3\oplus 0=3,\ 4\oplus 0=4,\ 5\oplus 0=5,\ 6\oplus 0=6\] Thus, 0 is the identity element. Hence, 0 is the identity element in the 7-hour clock addition. (d) When an element is added to its inverse, the result is the identity element. The identity element is 0. Find inverse of each element as, Let \[x\]be the inverse of any element, such that \[x\oplus 0=0\], and from the table \[0\oplus 0=0\] Thus, 0 is the inverse of element 0. \[x\oplus 1=0\], and from the table \[6\oplus 1=0\] Thus, 6 is the inverse of element 1. \[x\oplus 2=0\], and from the table \[5\oplus 2=0\] Thus, 5 is the inverse of element 2. \[x\oplus 3=0\], and from the table \[4\oplus 3=0\] Thus, 4 is the inverse of element 3. \[x\oplus 4=0\], and from the table \[3\oplus 4=0\] Thus, 3 is the inverse of element 4. \[x\oplus 5=0\], and from the table \[2\oplus 5=0\] Thus, 2 is the inverse of element 5. \[x\oplus 6=0\], and from the table \[1\oplus 6=0\] Thus, 1 is the inverse of element 6. Hence, inverse of elements 0,1,2,3,4,5 and 6 are 0,6,5,4,3,2 and 1, respectively. (e) \[4\oplus 5=2\] Now, find RHS as, \[5\oplus 4=2\] And, \[6\oplus 1=0\] Now, find RHS as, \[1\oplus 6=0\] Both the equations follow commutative property. Hence,\[4\oplus 5=5\oplus 4\]and\[6\oplus 1=1\oplus 6\]follows commutative property and has been verified.
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