Exercises
Exercise
2.19
An element \(l\) of a magma \(M\) is called a left identity if, for all \(m\in M\), \(lm=m\). Similarly, an element \(r\) of a magma \(M\) is called a right identity if, for all \(m\in M\), \(mr=m\). Suppose \(M\) is a magma having a left identity \(l\) and a right identity \(r\). Show that \(l=r\) and that \(l\) is the identity element of the magma.
Answer.
We have
\begin{align*} l & = lr \quad \text{as $r$ is a right identity element}\\ & = r \quad \text{as $l$ is a left idenitity element.} \end{align*}
Then, for any \(m\in M\), \(lm = m\) and \(ml = mr = m\). So \(l\) is the identity element of the magma.
Exercise
2.20
The cross product on \(\mathbb {R}^3\) is the binary operation given by
\[ (x_1,y_1,z_1)\times (x_2,y_2,z_2)= (y_1z_2-y_2z_1,z_1x_2-z_3x_1, x_1y_2-x_2y_1). \]
Show that the cross product is neither associative nor commutative. Then show that it has no identity element.