Dipper Operation/Examples/(m, n) = (3, 4)/Examples

Examples of Equations on Dipper Operation $+_{3, 4}$

Let $\N_{<7}$ denote the initial segment of the natural numbers:

$\N_{<7} := \set {0, 1, \ldots, 6}$

Let $+_{3, 4}$ be the dipper operation on $\N_{<7}$ defined as:

$\forall a, b \in \N_{<7}: a +_{3, 4} b = \begin{cases}

a + b & : a + b < 3 \\ a + b - 4 k & : a + b \ge 3 \end{cases}$

where $k$ is the largest integer satisfying:

$3 + 4 k \le a + b$

Example: $x +_{3, 4} 2 = 3$

Consider the equation:

$x +_{3, 4} 2 = 3$

This has the solutions:

$x = 1$

$x = 5$

Example: $x +_{3, 4} x = 4$

Consider the equation:

$x +_{3, 4} x = 4$

This has the solutions:

$x = 2$

$x = 4$

$x = 6$

Example: $x +_{3, 4} x = x$

Consider the equation:

$x +_{3, 4} x = x$

This has the solutions:

$x = 0$

$x = 4$