httpsproofwikiorgwikiIntegerMultiplicationWellDefined ProveS

https://proofwiki.org/wiki/Integer_Multiplication_Well-Defined

Prove.

Solution

>>>>>>>>>see note below >>>>>>

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LET ME USE F FOR PHI FOR EASE IN TYPING ....

F[N] = [ (1 , N+1) ] ...........HOPE YOU MEAN ORDERED PAIR ...ASSUMING SO

FOR N=A , WE GET .....F[A] = [ 1 , (A+1)]

FOR N=B , WE GET ....F[B] = [ 1 , (B+1) ]

F[A] + F[B ] = [ 2 , (A + B+2) ] .................................................1

N = [A+B] , WE GET .......F[A+B] = [ 1 , ( A+B+1) ] .............................2

HENCE ...F[A+B] IS NOT EQUAL TO F[A]+F[B] ...

PLEASE CHECK YOUR POST ....

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[F(A)] * [ F(B) ] = [1, ( A+1) ] * [ 1 , ( B+1) ] ..............MULTIPLICATION IS NOT DEFINED ....

WHAT SHALL WE TAKE IT AS ?????.........................................3

F[A*B] = [1 , (A*B+1) ] ............................................4

SO UNLESS MULTIPLICATION OF THE 2 ORDERED PAIRS IS DEFINED WE CAN NOT CHECK IF

F[A*B] = [ F(A) ] * [ F(B) ] .............PLEASE CHECK YOUR POST

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I FIND FROM THE LINK GIVEN THAT [(A,B)] HAS A TOTALLY DIFFERENT INTERPRETATION ,,,NOT THAT OF AN ORDERED PAIR ..
IT SHOWS CLOSOURE & DEFINES , ADDITION , MULTIPLICATION ETC..
AND THE ANSWER IS ALSO GIVEN ..THEN WHAT ELSE DO YOU WANT ..
PLEASE COME BACK WITH SPECIFIC DOUBTS IF ANY

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QUOTE

Let us define [[(a,b)]]? as in the formal definition of integers.

That is, [[(a,b)]]? is an equivalence class of ordered pairs of natural numbers under the congruence relation ?.

? is the congruence relation defined on N