I need help with this proof also

x = y <=> xz = yz

use only addition and multiplication for proof

x, y and z are natural numbers

thanks.

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- December 4th 2007, 06:14 AMjay0148Multiplication cancellation proof
I need help with this proof also

x = y <=> xz = yz

use only addition and multiplication for proof

x, y and z are natural numbers

thanks. - December 4th 2007, 10:24 AMJhevon
I do not see a problem here, you just want to go from the right to the left and then from the left to the right. Only basic algebraic manipulation is required.

Proof

Let

Suppose . Multiplying both sides by we obtain:

Thus we have

For the converse, suppose . Subtracting from both sides, we obtain: . Now , since , thus we must have . But this means .

Thus we have .

Therefore, if , .

QED - December 4th 2007, 06:53 PMjay0148
If anyone else can show this proof using only multiplication and addition, that would be helpful...thanks.