Let [a] = {x e Z | x and a are congruent mod m}
This is the exercise:
Describe the set [5] if m = 1. Show that [5] = [-1] in this case.
any IDeas?
To clarify did you mean,
$\displaystyle [a] = \{x \in \mathbb{Z} | x \equiv a \pmod m\}$
What part are you confused about?
x and a are congruent mod m means the remainder of $\displaystyle \frac{x}{m}$ is the same as the remainder of $\displaystyle \frac{a}{m}$
Because 5 mod 1 = 0, because $\displaystyle \frac{5}{1} = 5$ (with no remainder) we have,
[5] = {all integers that have a zero remainder when divided by 1} = {all integers} = $\displaystyle \mathbb{Z}$
Show that [-1] is the set of all integers too.