/sci/ - Science & Math

>>12525556
3. Associativity OK, so we need an identity and inverses. Fix any [math]a\in S[/math] and let [math]r_a, l_a\in S[/math] be such that [math]ar_a = a = l_aa[/math]. Given any [math]b\in S[/math], there exist [math]x, y\in S[/math] such that [math]ax = b = ya[/math], so now [math]l_ab = l_a(ax) = (l_aa)x = ax = b[/math] and, similarly, [math]br_a = b[/math]. Thus, those work as left and right identities, and this gives [math]r_a = l_ar_a = l_a[/math], from which it follows that [math]e := l_a[/math] is a mutual-sided identity. Next, consider the equations [math]ab = e, ca = e[/math]. These make sense by assumption. We have [math]b = eb = (ca)b = c(ab) = ce = c[/math], so any one-sided inverse is a mutual-sided inverse.
M.O.T.

>>11957446
Some are. My office mate is really cute and under different circumstances I would probably hang around his desk so that our h*nds could accidentally touch and mmhm. There was also a super cute guy in a conference earlier this year and I would have liked to make moves on him.

I hope you all had a nice maths week.