Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $*$ be a binary operation acting on a non-empty set $S$, such that $$(a*b)*a=b,$$ for all $a,b\in S$.

Prove that $$a*(b*a)=b,$$ for all $a,b \in S$.

share|improve this question
    
The solution in mathlinks says $((b*a)*b)(b*a)=b$ out of the blue. Why? –  Modded Bear Mar 21 '14 at 23:18
1  
Let $a=(b*a),b=b.$ –  coffeemath Mar 21 '14 at 23:22
    
what do you mean let $a=b*a$ –  Modded Bear Mar 21 '14 at 23:33
    
@coffeemath can we just go ahead and say $a=(b*a)$ is that legit? –  Modded Bear Mar 21 '14 at 23:43
    
user4140 Yes it's legit. To se it rewrite (ab)*a=b with new letters like (xy)*x=y, then let this x be (a*b) and this y be b. –  coffeemath Mar 22 '14 at 0:10

2 Answers 2

up vote 13 down vote accepted

We have that $$ a*(b*a)=\big((b*a)*b\big)*(b*a)=b $$ Note that the second "=" holds since $$ (c*b)*c=b, $$ where $c=b*a$.

share|improve this answer
    
yes, the only part I don't get is the second equivalence –  Modded Bear Mar 21 '14 at 23:33
1  
Why is $((b*a)*b)*(b*a)=b$??? –  Modded Bear Mar 21 '14 at 23:35
    
I really don't understand it. Could you elaborate please? –  Modded Bear Mar 21 '14 at 23:41
    
@user4140: See update. –  Yiorgos S. Smyrlis Mar 21 '14 at 23:47
    
Ohhhhhh, I finally get it. Thanks. –  Modded Bear Mar 22 '14 at 0:00

Because $b*a,b\in S$, so by the formula $((b*a)*b)*(b*a)=b$. Then $a*(b*a)=b$.

share|improve this answer
    
By what formula? –  Modded Bear Mar 21 '14 at 23:43
    
@user4140 By the formula $a∗(b∗a)=b$, just replace $a$ by $a*b$. –  user2345215 Mar 21 '14 at 23:47

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.