# Thread: need help with equivalence class

1. ## need help with equivalence class

For x belonging to Z, let us denote by [x] equivalence class which contains x. Prove that if [x]=[x'] and [y]=[y'] then [x+y]=[x'+y'] and [xy]=[x'y']
Remark: When xRy people say that "x and y are congruent mod 3", and sometimes write x equivalent y (mod3)

someone can help me please here???

2. Originally Posted by tukilala
For x belonging to Z, let us denote by [x] equivalence class which contains x. Prove that if [x]=[x'] and [y]=[y'] then [x+y]=[x'+y'] and [xy]=[x'y']
Remark: When xRy people say that "x and y are congruent mod 3", and sometimes write x equivalent y (mod3)

someone can help me please here???
If $\displaystyle [x]$ denotes the equivalence class in $\displaystyle \mathbb{Z}$ modulo $\displaystyle 3$ then all elements of $\displaystyle [x]$ have the same remainder as $\displaystyle x$ when divided by $\displaystyle 3$, and any integer which has this remainder is in $\displaystyle [x]$.

So:

$\displaystyle [x]=[x']$ and $\displaystyle [y]=[y']$ means there are integers $\displaystyle a,\ b,\ d,\ e,\ r_1,\ r_2$ where $\displaystyle r_1,\ r_2 \in \{0,1,2\}$ such that:

$\displaystyle x=3a+r_1,\ x'=3b+r_1,\ y=3d+r_2,\ y'=3e+r_2$

Then $\displaystyle x+y=3(a+d)+(r_1+r_2)$, and so $\displaystyle [x+y]=[r_1+r_2]$

Similarly:

$\displaystyle x'+y'=3(b+e)+(r_1+r_2)$, and so $\displaystyle [x'+y']=[r_1+r_2]$

Hence: $\displaystyle [x+y]=[x'+y']$

CB