Defining & Proving Modulo Multiplication in $\mathbb{Z}_k$

[mathmari]

Hey!

Based on the definition of $+_k$, I want to give an inductive definition for the multiplication $\cdot_k$ in $\mathbb{Z}_k$, such that for all $x\in \mathbb{N}_0$ and $y\in \mathbb{N}_0$ it holds that $$x\cdot_k y=(x\cdot y)\mod k$$

What is an inductive definition? (Wondering)

Do we write $x\cdot y=y+y+\ldots +y$ ($n$-times) and we apply each time the addition $+_k$ ? (Wondering)
I want to prove also by induction that for a $x\in \mathbb{N}_0$ it holds for all $y\in \mathbb{N}_0$ that $$(x\cdot y)\mod k=(x\mod k)\cdot_k (y\mod k)$$

We apply the induction on $y$ or not? (Wondering)

 

[mathmari]

I want to prove also by induction that for a $x\in \mathbb{N}_0$ it holds for all $y\in \mathbb{N}_0$ that $$(x\cdot y)\mod k=(x\mod k)\cdot_k (y\mod k)$$

We apply the induction on $y$ or not? (Wondering)

Is the induction as follows? Base case: $y=1$ : $$(x\cdot 1)\mod k=x\mod k$$ The other side is $$ (x\mod k)\cdot_k (1\mod k)=(x\mod k)\cdot_k 1=x\mod k$$ So, they are equal. Inductive Hypothesis: We suppose that it holds for $y=n$ : $$(x\cdot n)\mod k=(x\mod k)\cdot_k(n\mod k)$$ Inductive step: We will show that it holds for $y=n+1$ : $$(x\cdot (n+1))\mod k=(x\cdot n+x)\mod k=(x\cdot n)\mod k+x\mod k \ \ \overset{\text{ Inductive Hypothesis } }{ = } \ \ (x\mod k)\cdot_k (n\mod k)+x\mod k$$ The other side is $$(x\mod k)\cdot_k ((n+1)\mod k)=(x\mod k)\cdot_k (n\mod k+1)\\ =(x\mod k)\cdot_k (n\mod k)+x\mod k$$ So, they are equal.

Is this correct? Could I improve something? (Wondering)

 

[Evgeny.Makarov]

What is an inductive definition?

Is there a reason why you can't look this up in your lecture notes or textbook? Or see Wikipedia.

Do we write $x\cdot y=y+y+\ldots +y$ ($n$-times) and we apply each time the addition $+_k$ ?

No inductive definition uses ellipsis.

Is the induction as follows?

Until there is a definition, it makes no sense to prove anything.

 

[mathmari]

An inductive definition for the multiplication $\cdot_k$ in $\mathbb{Z}_k$ is the following:
\begin{align*}&x\cdot_k 0=0 \\ &x\cdot_k (y+1)=x\cdot_k y+_kx, \ y\geq 0\end{align*}
right? (Wondering)