jkwiens.com: Question: Binary operation Problem

Question: Binary operation Problem

Sunday, 13 of January , 2008 at 5:22 pm

This weekend I dropped by the UCFV library and became a community member. The library doesn’t have the biggest selection, but it is cheaper than buying textbooks. Regardless, I picked up a textbook on Abstract Algebra. I didn’t get an opportunity to take an Abstract Algebra course at university, so I thought it would be a fun topic to learn.

Question: Assume that a binary operation $\Box$ on a set has a left unit and satisfies the identity $x \Box (y \Box z) = (x \Box z) \Box y$ $\forall x,y,z \in X$ . Prove that $\Box$ is associative and commutative.

Definitions:
Left Unit: $x = u \Box x$ $\forall x \in X$ where $u \in X$
Commutative: $x \Box y = y \Box x$
Associative: $x \Box ( y \Box z)= (x \Box y) \Box z$

Category: Algebra, Questions

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