Return to Homework 3, Glossary, Theorems

### Problem 4

(*) An identity element for a binary operation $∗$ as described in class is sometimes called a “two-sided

identity element”. Using complete sentences, give analogous definitions for

(a) a left identity element $e_{L}$ for $∗$, and

(b) a right identity element $e_{R}$ for $∗$.

Theorem 3.13 shows that if a two-sided identity element for $∗$ exists, it is unique. Is the same true for

a one-sided identity element you just defined? If so, prove it. If not, give a counter example $\langle S, *\rangle$ for

a finite set $S$ and find the first place where the proof of Theorem 3.13 breaks down.

**Solution**

(a) Let $*$ be a binary operation on a set $S$. An element $e_{L} \in S$ is a left identity element for $*$ if and only if $e_{L} * s = s$ for all $s \in S$.

(b) Let $*$ be a binary operation on a set $S$. An element $e_{R} \in S$ is a right identity element for $*$ if and only if $s * e_{R} = s$ for all $s \in S$.

Let $*$ be a binary operation on $S$ defined by $a * b = a$ for all $a, b \in S$. Then for all $b \in S$, $b$ is a right identity element, and right identity elements are not unique. Similarly, left identity elements are not unique. In the proof of Theorem 3.13, the first incorrect statement would be that, using $\bar{e}_{L}$ as a left identity, $e_{L} * \bar{e}_{L} = e_{L}$.