Answer to Question #350786 in Abstract Algebra for Eliz T.

Question

2.1. Let S be any set. Prove that the law of multiplication defined

by ab = a is associative.

Answer 1

Let $x,y,z\in S$. We want to show that $x(yz)=(xy)z$.

Indeed $x(yz)=xy=x$ by the law of multiplication in $S$. And $(xy)z=xz=x$ by the same law so $x(yz)=x=(xy)z$.