jkwiens.com: Answer: Binary operation Problem

Answer: Binary operation Problem

Saturday, 19 of January , 2008 at 2:17 pm

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.

Answer - Proving that $\Box$ is commutative:
First, let x be equal to the left unit, . Since the identity $x \Box (y \Box z) = (x \Box z) \Box y$ is true $\forall x \in X$ and since $u \in X$ because of the definition of a left unit, therefore $u \Box (y \Box z) = (u \Box z) \Box y$ .

Since $z \in X$ , which will mean $z = u \Box z$ and therefore

$u \Box (y \Box z) = (u \Box z) \Box y \Longrightarrow u \Box (y \Box z) = z \Box y$

Because $\Box$ is a binary operation, therefore $\Box : X \times X \rightarrow X$ . This means that $y \Box z \in X$ . Therefore, we know from the definition of a left unit that $u \Box (y \Box z) = y \Box z$ .

This means:

$u \Box (y \Box z) = z \Box y \Longrightarrow y \Box z = z \Box y$

$\therefore \Box$ is commutative.

Answer - Proving that $\Box$ is associative:
Since the identity $x \Box (y \Box z) = (x \Box z) \Box y$ is true $\forall x,y,z \in X$ , we can rename the variables while maintaining the validity of the identity.

$\therefore x \Box ( z \Box y ) = ( x \Box y) \Box z$ $\forall x,y,z \in X$

If we use the commutative law (which we just proved), we know that $z \Box y = y \Box z$ .

$\therefore x \Box ( z \Box y ) = ( x \Box y) \Box z \Longrightarrow x \Box ( y \Box z) = ( x \Box y) \Box z$

This means $\Box$ is associative.

Category: Algebra, Answers

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