Cancellation Laws hold in a group proof (Abstract Algebra)

[@mikeCavalle]

btw - i argue that proofwiki.org/wiki/Cancellation_Laws is also incorrect reasoning. i still say that given ab=ac ==> a'ab = a'ac is using Cancellation since we get that a'ab =a'ac ==> ab = ac. Until you prove cancellation you can't use "multiple on the left on both sides of =" as an operation. If the symbol "equal(=)" is describing the notion of identity and is taken to be "is" then getting a'ab is a'ac from ab is ac is using the Cancellation Law cause identity is symmetric .

[@thapelo_inori]

you made that look easy. Because it is. Thanks ; )

[@BriTheMathGuy]

You're welcome! Have a nice day!

[@muhammadrasull9388]

@@BriTheMathGuy Thanks for you

[@AlexandreGurchumelia]

0:53 you should reference associativity here

[@mikeCavalle]

your are using the cancellation law to prove the cancellation law. ab = ac ==> a"ab = a''ac (ie. multiplying on both sides) assumes that you already have a cancellation law. right? when proving a cancellation law all you have is the Group definition,right? .

[@mikeCavalle]

btw : Denote the unit of the group by e then: ab=ac implies: b=eb=(a−1a)b=a−1(ab)=a−1(ac)=(a−1a)c=ec=c ba=ca implies: b=be=b(aa−1)=(ba)a−1=(ca)a−1=c(aa−1)=ce=c Note that unit, inverse and associativity (characteristics of group) all play a part.

[@BriTheMathGuy]

Im using associativity, though I didn’t explicitly say so. You may verify this proof by other sources. Here is one for you to check out: proofwiki.org/wiki/Cancellation_Laws Have a wonderful day!

[@VinayKumar-nf6sd]

@@mikeCavalle He is assumed to have proven that the identity element in a group is unique

[@mikeCavalle]

where in the the definition of a Group does it say you can multiply both sides of an equation by the same thing and maintain equality?

[@hrcricket3240]

Is the converse true ? Please ans and justify your answer

[@AkashPatel-Sky]

Thank u

[@BriTheMathGuy]

You’re welcome, have a nice day!

[@AkashPatel-Sky]

@@BriTheMathGuy glad to have a reply from u sir...love from India

[@ibrahimfarhanparinduri1276]

Can we cancel the a if the equation is ab=ya?

[@saifmiziry6457]

thanks sir , but now i want to sol my quastion : q/ (In agroup theory , the cancellation is hold) . is this quastion looks like your quastion or not ?

[@cardiacbalakumaranm5420]

Thanks sir

[@AngelaEstherRS]

What happens if you use numbers?

[@BriTheMathGuy]

You will get the same result, though it would not be a complete proof.

[@cloudwithjd4068]

Good explanation sir Tq

[@BriTheMathGuy]

You're welcome!

[@mahdihasan3145]

thnx sir

[@BriTheMathGuy]

You're welcome!