MHB -412.7.3 decide if cosets of H are the same

[karush]

Let $H=\{0;\pm 3;\pm 6;\pm 9;\cdots\}$.
$\textit{ Use }$
$$aH=bH \textit{ or }aH\cap bH=\oslash$$
$\textit{then..}$
$$aH = bH \textit{ iff } (b-a) \textit{ is in } \textit{H}$$
to decide whether or not the following cosets of H are the same.
$\textsf{a. 11 + H and 17 + H}$
$\textsf{b. -1 + H and 5 + H}$
$\textsf{c. 7 + H and 23 + H}$ok not sure what the official method of this would be but for a. 11+6=17 so the coset would would be just a shift over 6 places. the same shift seens to be true for b and c however the beginning numbers would be different

however the book says that c is no

 

[Opalg]

Let $H=\{0;\pm 3;\pm 6;\pm 9;\cdots\}$.
$\textit{ Use }$
$$aH=bH \textit{ or }aH\cap bH=\oslash$$
$\textit{then..}$
$$aH = bH \textit{ iff } (b-a) \textit{ is in } \textit{H}$$
to decide whether or not the following cosets of H are the same.
$\textsf{a. 11 + H and 17 + H}$
$\textsf{b. -1 + H and 5 + H}$
$\textsf{c. 7 + H and 23 + H}$ok not sure what the official method of this would be but for a. 11+6=17 so the coset would would be just a shift over 6 places. the same shift seens to be true for b and c however the beginning numbers would be different

however the book says that c is no

$H = \{\ldots -9;-6;-3;\ 0;\ 3;\ 6;\ 9 \ldots\}$

For a.: $11+H = \{\ldots 2;\ 5;\ 8;\ 11;\ 14;\ 17;\ 20;\ \ldots\},\quad 17+H = \{\ldots 8;\ 11;\ 14;\ 17;\ 20;\ 23;\ 25 \ldots\}$
Those two sets are the same. The numbers in the list have just been shifted by two places to get from $11+H$ to $17+H$.

For c.: $7+H = \{\ldots -2;\ 1;\ 4;\ 7;\ 10;\ 13;\ 16;\ 19;\ 22;\ 25;\ 28 \ldots\},\quad 23+H = \{\ldots 14;\ 17;\ 20;\ 23;\ 26 \ldots\}$
Those two sets are not the same. The numbers in them do not coincide at all.

Can you see what condition the shifts must satisfy in order for the two cosets to be the same?

 

[HOI]

Yes: your "method" works. 11+ H and 17+ H are the same because 17- 11= 6 is a multiple of 3; -1+ H and 5+ H are the same because 5- (-1)= 6 is a multiple of 3; and 7+ H and 23+ H are NOT the same because 23- 7= 16 is NOT a multiple of 3.

 

[karush]

thanks everyone
that was a great help

I'll post more for sure

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