Inverse Element/Examples
Examples of Inverse Elements
Addition Modulo $6$
Consider the additive group of integers modulo $6$, whose Cayley table is given below:
$\quad \begin{array}{r|rrrrrr}
\struct {\Z_6, +_6} & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\
\hline
\eqclass 0 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6
\\
\eqclass 1 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6
\\
\eqclass 2 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6
\\
\eqclass 3 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6
\\
\eqclass 4 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6
\\
\eqclass 5 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6
\\
\end{array}$
Each element of this group is invertible.
Multiplication Modulo $6$
Consider the multiplicative monoid of integers modulo $6$ , whose Cayley table is given below:
$\quad \begin{array} {r|rrrrrr}
\struct {\Z_6, \times_6} & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\
\hline
\eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6
\\
\eqclass 1 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6
\\
\eqclass 2 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6
\\
\eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6
\\
\eqclass 4 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6
\\
\eqclass 5 6 & \eqclass 0 6 & \eqclass 5 6 & \eqclass 4 6 & \eqclass 3 6 & \eqclass 2 6 & \eqclass 1 6
\end{array}$
The only invertible elements of this group are $\eqclass 1 6$ and $\eqclass 5 6$.
Symmetry Group of Square
Consider the symmetry group of the square:
Let $\SS = ABCD$ be a square.
The various symmetry mappings of $\SS$ are:
- the identity mapping $e$
- the rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ around the center of $\SS$ anticlockwise respectively
- the reflections $t_x$ and $t_y$ are reflections in the $x$ and $y$ axis respectively
- the reflection $t_{AC}$ in the diagonal through vertices $A$ and $C$
- the reflection $t_{BD}$ in the diagonal through vertices $B$ and $D$.
This group is known as the symmetry group of the square, and can be denoted $D_4$.
Each element of this group is invertible, for example:
- $r^{-1} = r^3$
- ${t_x}^{-1} = t_x$
Rational Multiplication
Consider the multiplicative group of positive rational numbers $\struct {\Q_{> 0}, \times}$.
- $\tfrac 4 {23}$ and $5 \tfrac 3 4$ are inverses of each other.
Square Root Function
Let $\struct {\CC, \circ}$ be the monoid of all real functions $\CC$ under composition $\circ$ over the closed real interval $\closedint 0 1$.
Not all elements of $\CC$ have an inverse mapping, but in particular let $f$ be defined as:
- $\forall x \in \closedint 0 1: \map f x = x^2$
$\map f x$ is in fact a bijection and has inverse mapping $\inv f x$ defined as:
- $\forall x \in \closedint 0 1: \inv f x = \sqrt x$