43 votes 43 votes 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? It is not closed $2$ does not have an inverse $3$ does not have an inverse $8$ does not have an inverse Set Theory & Algebra gatecse-2006 set-theory&algebra group-theory normal + – Rucha Shelke asked Sep 16, 2014 Rucha Shelke 9.9k views answer comment Share Follow See all 7 Comments See all 7 7 Comments reply Show 4 previous comments HitechGa commented Jul 14, 2021 reply Follow Share While solving this question for the first time, I had a weird feeling about the meaning and options of the questions. The definition of group which I had in the back of my mind, rather the algorithm which I use to check whether a set and a binary operation is a group or not is as follows: A monoid with identity element $e$ is a group, iff for each element $a$ in the monoid, there exists as element $x$ such that $a*x=x*a=e$. So $x$ is the inverse of $a$ i.e. $x=a^{-1}$. So intuitively, in the process of checking, if a structure failed to be an algebraic structure [closure property not followed], then it is not a group obviously. And this step-by-step checking of structures made me feel that $(A)$ is the only true option for the given structure to be not a group, Rest all are false. [Hence I thought correct options are BCD] But the alternate definition of groups (in texts which talks about groups directly without talking about algebraic structure, semi group, monoids) says: A set $S$ with a binary operation $*$ defined on its elements is a group iff (1) $S$ is closed under $*$ and (2) $*$ is associative and (3) there exists an identity element and (4) There exists inverse for each element. From this alternative definition, since the requirements for being a group is given in AND form, then for NOT being a group the requirements shall be : $F = \lnot(1) \lor \lnot(2) \lor \lnot(3) \lor \lnot(4)$ Although if anyone of $(1)$ or $(2)$ or $(3)$ or $(4)$ is false then $F$ shall be true, but strictly speaking there is no precedence or order among the terms of $F$. They can just be rearranged and hence options $B$ and $D$ are also true since make $\lnot(4)$ true. 5 votes 5 votes Akash 15 commented Dec 27, 2023 reply Follow Share 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: 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. $2$ 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. $3$ doesn't have an inverse No. $3$ has an inverse, which is $7$ $(3\times 7)mod\;10=1;\;\;(7\times 3)mod\;10=1\;\;\therefore7=3^{-1}$ $8$ 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. $\color{DarkGreen} Only\;C\;is\;false$ 1 votes 1 votes Bhaskar_Saini commented Apr 6 reply Follow Share 5*8 = 40 Mod 10 = 0, which is not in the group. It is not closed. 0 votes 0 votes Please log in or register to add a comment.
Best answer 33 votes 33 votes Answer: C $3$ has an inverse, which is $7.$ $3*7 \mod 10 = 1.$ Rajarshi Sarkar answered May 11, 2015 • selected Jun 2, 2015 by Rajarshi Sarkar Rajarshi Sarkar comment Share Follow See all 9 Comments See all 9 9 Comments reply shikharV commented Dec 28, 2015 reply Follow Share The given relation is not closed also, 2*8 mod 10 = 6 which doesn't belong to the set. Can it be a reason for it not being a group? 1 votes 1 votes Himanshu1 commented Dec 28, 2015 reply Follow Share yes , it is one of the reasons. 7 votes 7 votes mehul vaidya commented Jul 13, 2019 reply Follow Share also 5*2 mod 10 =0 not in list 2 votes 2 votes vikimsh commented Jan 9, 2020 reply Follow Share also 8*5 mod 10 =0 not in list. 2 votes 2 votes raja11sep commented Nov 3, 2021 reply Follow Share also 2*3 mod 10 =6 not in list. 1 votes 1 votes Shoto commented Jan 22, 2022 reply Follow Share also 2*2 mod 10 = 4 not in list 5 votes 5 votes raja11sep commented Oct 18, 2022 reply Follow Share also 8*8 mod 10 = 4 not in list 0 votes 0 votes Pranavpurkar commented Oct 21, 2022 reply Follow Share LoL! 0 votes 0 votes Akash 15 commented Dec 27, 2023 i edited by Akash 15 Dec 27, 2023 reply Follow Share 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 0 votes 0 votes Please log in or register to add a comment.
4 votes 4 votes Ans C. Vikrant Singh answered Dec 31, 2014 Vikrant Singh comment Share Follow See all 2 Comments See all 2 2 Comments reply anshu commented Feb 4, 2015 reply Follow Share How?? 0 votes 0 votes Sandeep Suri commented Jan 11, 2017 reply Follow Share 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 1 votes 1 votes Please log in or register to add a comment.
3 votes 3 votes 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 . worst_engineer answered Jul 18, 2015 worst_engineer comment Share Follow See all 3 Comments See all 3 3 Comments reply vishal8492 commented Jul 20, 2015 reply Follow Share 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. 8 votes 8 votes worst_engineer commented Jul 20, 2015 reply Follow Share Ohh yeah.. sorry didn't see the question properly. my bad :( . Yes , it is true. thanks 0 votes 0 votes Sandeep Suri commented Jan 9, 2018 reply Follow Share Question is asking which of the statement is not FALSE.. Yes it is not closed is a True statement. 0 votes 0 votes Please log in or register to add a comment.
0 votes 0 votes . akshay_123 answered Sep 2, 2023 akshay_123 comment Share Follow See all 0 reply Please log in or register to add a comment.