GATE CSE 2006 | Question: 3

Question

The set $\{1,2,3,5,7,8,9\}$ under multiplication modulo $10$ is not a group. Given below are four possible reasons. Which one of them is false?

• A. It is not closed
• B. $2$ does not have an inverse
• C. $3$ does not have an inverse
• D. $8$ does not have an inverse

Comments

Given set $S=\{1,2,3,5,7,8,9\}$          $(\{1,2,3,5,7,8,9\},\bigotimes_{10})$ is not a group.

Option by option:

1. It is not closed

$(2\times 3)  mod \;10=6 \notin S$

$(2\times 5)  mod \;10=0 \notin S$

$(2\times 2)  mod \;10=4 \notin S$

So we can observe $\{0,4,6\}$ these are not in $S$. This group is not closed as they not belong to our base set.

2. $2$ doesn’t have an inverse

$(2\times 1)  mod \;10=2 \in S$

$(2\times 2)  mod \;10=4 \notin S$

$(2\times 3)  mod \;10=6 \notin S$

$(2\times 5)  mod \;10=0 \notin S$

$(2\times 7)  mod \;10=4 \notin S$

$(2\times 8)  mod \;10=6 \notin S$

$(2\times 9) mod\;10=8 \in S$

So, here is no identity element. Thus the inverse of $2$ not even exist.

100. $3$ doesn't have an inverse

No. $3$ has an inverse, which is $7$ 

$(3\times 7)mod\;10=1;\;\;(7\times 3)mod\;10=1\;\;\therefore7=3^{-1}$

500. $8$ doesn’t have an inverse

$(8\times 1)  mod \;10=8 \in S$

$(8\times 2)  mod \;10=6 \notin S$

$(8\times 3)  mod \;10=4 \notin S$

$(8\times 5)  mod \;10=0 \notin S$

$(8\times 7)  mod \;10=6 \notin S$

$(8\times 8)  mod \;10=4 \notin S$

$(8\times 9) mod\;10=2 \in S$

So, here is no identity element. Thus the inverse of $8$ not even exist.

$\color{DarkGreen} Only\;C\;is\;false$ 

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5*8 = 40 Mod 10 = 0, which is not in the group. It is not closed.

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question is mindblowing only because of the framing that the options are true and asked for false stmt .But in hurry none cares

Answer 1

Answer: C

$3$ has an inverse, which is $7.$
$3*7 \mod 10 = 1.$

Comments

also 8*8 mod 10 = 4 not in list

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LoL!

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also 7*8 mod 10 = 6 not in list

Follow this trend :)

If you want option analysis then you can look this:

https://gateoverflow.in/882/gate-cse-2006-question-3?show=417093#c417093

Answer 2

Ans C.

Comments

How??

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In the question it is asked which is false. and given that it's not a group. As we can see 3 have an inverse which is 7 therefore this statement is false

Answer 3

Hi , please correct me if I am wrong. The set is {1,2,3,5,7,8,9} and we need to do multiplication modulo 10.  So, 2 (multiplication modulo 10 ) 2 = 4 , which is not in the set. That means , it is not closed.

Please correct me , if I am wrong .

Comments

You're right 0,4,6 are missing ; so for every n mod 10 = {0,4,6} this set is definitely not closed.

But trick is it is not closed is not false reason ; it's True.

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Ohh yeah.. sorry didn't see the question properly. my bad :( . Yes , it is true. thanks

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Question is asking which of the statement is not FALSE.. Yes it is not closed is a True statement.

Answer 4

$\begin{array}{|c|ccccccc|} \hline
⊗_{10}&1&2&3&5&7&8&9\\ \hline
1&1&2&3&5&7&8&9\\ \hline2&2&4&6&0&4&6&8\\ \hline3&3&6&9&5&1&4&7&\\ \hline5&5&0&5&5&5&0&5&\\ \hline7&7&4&1&5&9&6&3&\\ \hline8&8&6&4&0&6&4&2&\\\hline9&9&8&7&5&3&2&1\\ \hline
\end{array}$

from the above composition table, it is clear that $1$ is the identity element.

1. it is not closed: true, because elements like $0,4,6\notin$ the set $\left \{ 1,2,3,5,7,8,9 \right \}$
2. $2$ doesn't have an inverse: true
3. $3$ does not have an inverse: false, inverse (3)=7
4. $8$ does not have an inverse: true.

So Option $(C)$ is the correct.

Comments

nice one

Answer 5

.

Answer 6

Why option A is correct ?? we know possible remainder when divide by n is 0,1,2,3,4 .... n-1 which is not in Set S so it is not  covering all remainders properly.