Well-Defined Mapping/Examples/Floor of Half Function on Congruence Modulo 6

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Example of Mapping which is not Well-Defined

Let $x \mathrel {C_6} y$ be the equivalence relation defined on the natural numbers as congruence modulo $6$:

$x \mathrel {C_6} y \iff x \equiv y \pmod 6$

defined as:

$\forall x, y \in \N: x \equiv y \pmod 6 \iff \exists k, l, m \in \N, m < 6: 6 k + m = x \text { and } 6 l + m = y$

Let $\eqclass x {C_6}$ denote the equivalence class of $x$ under $C_6$.

Let $\N / {C_6}$ denote the quotient set of $\N$ by $C_6$.


Let us define the mapping $\phi$ on $\N / {C_6}$ as follows:

$\map \phi {\eqclass x {C_6} } = \eqclass {\floor {x / 2} } {C_6}$

where $\floor {\, \cdot \,}$ denotes the floor function.


Then $\phi$ is not a well-defined mapping.


Proof

Consider $x = 4$ and $x' = 10$.

We have that:

$4 = 6 \times 0 + 4$

and:

$10 = 6 \times 1 + 4$

and so:

$4 \mathrel {C_6} {10}$

That is:

$\eqclass 4 {C_6} = \eqclass {10} {C_6}$


However, we have that:

$\map \phi {\eqclass 4 {C_6} } = \eqclass 2 {C_6}$

while:

$\map \phi {\eqclass {10} {C_6} } = \eqclass 5 {C_6}$

But $\eqclass 2 {C_6} \ne \eqclass 5 {C_6}$

Hence we have $x, x' \in \N$ for which it is not the case that:

$\map \phi {\eqclass x {C_6} } = \map \phi {\eqclass {x'} {C_6} }$

That is, $\phi$ is not well-defined.

$\blacksquare$


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