Multiplication -- prove this one by induction

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Homework Help Overview

The discussion revolves around proving a property of multiplication using mathematical induction, specifically focusing on the expression n x (m++) = (n x m) + n for natural numbers n and m. The original poster (OP) attempts to establish a base case and a recursive step but encounters difficulties in proving the equivalence of both sides in the induction step.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants question the notation m++, with some suggesting it may not be standard mathematical terminology and could be interpreted differently. The OP's approach to induction is noted, but clarity on the notation is sought. There are discussions about whether m++ is shorthand for m + 1 and how this impacts the proof.

Discussion Status

The discussion is ongoing, with participants providing insights into the notation and its implications for the proof. Some guidance is offered regarding the use of standard arithmetic axioms, and there is a suggestion to clarify the axioms available for the proof. Multiple interpretations of the notation are being explored without reaching a consensus.

Contextual Notes

There is a lack of clarity regarding the notation m++, which leads to questions about its meaning and implications for the proof. Participants are also considering the axioms that can be used in the proof, indicating potential constraints in the approach.

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Homework Statement
Prove by induction that for any natural numbers n and m , n x (m++)= (n x m) + n
Relevant Equations
n x (m++) = (n x m) +n
Prove by induction that for any natural numbers n and m , n x (m++)= (n x m) + n

The base case, n=0 gives 0 x m++=(0 x m) +0 gives 0=0
Now assume n x (m++) = (n x m) +n
For n++ we get

n++(m++)=((n++)m) + n++

from this point I am stuck, how can I prove both sides are the same?
 
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What do you mean by m++ ? In mathematics, this means nothing. Do you mean the programming language C convention? But then, actually n * (m++) == n *m is true, while n * (++m) == n * m + n is true.
 
PAllen said:
What do you mean by m++ ? In mathematics, this means nothing. Do you mean the programming language C convention? But then, actually n * (m++) == n *m is true, while n * (++m) == n * m + n is true.
The OP posted a similar question using this notation in another thread (https://www.physicsforums.com/threads/proof-multiplication-is-commutative.782057/#post-6537515).

I believe it is less sophisticated notation than is used in C et al, and has nothing to do with pre- or post-increment. I believe that the notation m++ is just shorthand for m + 1.
 
Mark44 said:
The OP posted a similar question using this notation in another thread (https://www.physicsforums.com/threads/proof-multiplication-is-commutative.782057/#post-6537515).

I believe it is less sophisticated notation than is used in C et al, and has nothing to do with pre- or post-increment. I believe that the notation m++ is just shorthand for m + 1.
shorthand in what way? Same number of characters, and + needs right pinky reach as well as left pinky shift, while 1 just needs left pinky reach (at least on my keyboard).
 
PAllen said:
shorthand in what way?
OK, maybe samehand...
 
Can you say what axioms you are allowed to use? I would also recommend explicitly writing + 1 to better use normal arithmetic axioms.
 
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