MHB Induction on the number of equations

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The discussion focuses on proving a statement about differential operators and systems of equations using mathematical induction. It establishes a base case for one equation and an inductive hypothesis for k equations, asserting that they can be expressed in a specific form. The inductive step aims to demonstrate that if the hypothesis holds for k equations, it also holds for k+1 equations by combining the equations to derive a new differential equation. The participants seek clarity on the correctness of their reasoning and how to proceed with the proof. Overall, the thread emphasizes the structured approach of induction in mathematical proofs related to differential equations.
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Hey! :o

Let $L$ be a differential operator.

We suppose that we have $n$ equations, that means $\phi: \displaystyle{\bigwedge_{j=1}^n L_j x=f_j}$ and we assume that $\phi$ can be written as $Lx=f \land \psi$, where $\psi$ doesn't contain any $x$.

We prove this by induction on the number of equations, $n$.

  • Base case: For $n=1$ we have one equation, so it is of the form $Lx=f$.
  • Inductive hypothesis: We suppose that it holds for $n=k$, i.e., if $\phi$ contains $k$ equations, then we can reduce it into the form $$Lx=f \land \psi \ \ \text{ where } \psi \text{ doesn't contain any } x.$$
  • Inductive step: We will show that it holds for $n=k+1$, i.e., if we have $k+1$ equations we can reduce this system into the form $$Lx=f \land \psi \ \ \text{ where } \psi \text{ doesn't contain any } x.$$
    From the inductive hypothesis we know that we can reduce the first $k$ equations into the above form. So we have two equations that contain $x$ and its derivatives.

Is this correct so far? (Wondering)

How could we continue? (Wondering)
 
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Is the inductive step as follows? (Wondering) Inductive step: We will show that it holds for $n=k+1$, i.e., if we have $k+1$ equations we can reduce this system into the form $$Lx=f \land \psi \ \ \text{ where } \psi \text{ doesn't contain any } x.$$
From the inductive hypothesis we know that we can reduce the first $k$ equations into the above form. So we have two equations that contain $x$ and its derivatives. We add these two equations and we get a new differential equation $Lx=f$. So the initial system is equivalent to the new differential equation $Lx=f$.
 
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I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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