- #1

Von Neumann

- 101

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- TL;DR Summary
- Based on the properties provided, why does x + 3 = 5 lead to (x + 3) + (-3) = 5 + (-3)?

I am 100% reading too much into this, but I am curious which of the properties provided by Spivak allow one to justify a specific argument. For reference/context, the properties are:

P1: If a, b, and c are any numbers, then

$$a +(b + c) = (a + b) +c$$

P2: If a is any number, then

$$a + 0 = 0 + a = a$$

P3: For every number a, there is a number -a such that

$$a + (-a) = (-a) + a = 0$$

Specifically, I am curious about the following:

$$ x + 3 = 5 $$

$$ x + 3 + (-3) = 5 + (-3) $$

From this point, I understand the argument as the following:

By Property P3,

$$ x + 0 = 2 $$

By Property P2 ,then

$$ x = 2 $$

Is the step in question justifiable from any of the listed properties? Or, rather, is this basic property of addition intended to be assumed? Most properties are introduced from first principles, so I don't know if this should be any different. However, directly before the proof Spivak says "It is then possible to find the solution of certain simple equations by a series of steps (each justified by P1, P2, or P3) ...".

Here is the best that I can come up with:

$$x + 3 = 5$$

By P2,

$$x + 3 + 0 = 5 $$

By P3,

$$x + 3 + (-3 + 3) = 5 $$

By P1,

$$x + (3 + (-3)) + 3 = 5$$

Then, (by basic algebra?)

$$x + (3 + (-3)) = 5 + (-3)$$

By P3,

$$x + 0 = 2$$

By P2,

$$x = 2$$

P1: If a, b, and c are any numbers, then

$$a +(b + c) = (a + b) +c$$

P2: If a is any number, then

$$a + 0 = 0 + a = a$$

P3: For every number a, there is a number -a such that

$$a + (-a) = (-a) + a = 0$$

Specifically, I am curious about the following:

$$ x + 3 = 5 $$

$$ x + 3 + (-3) = 5 + (-3) $$

From this point, I understand the argument as the following:

By Property P3,

$$ x + 0 = 2 $$

By Property P2 ,then

$$ x = 2 $$

Is the step in question justifiable from any of the listed properties? Or, rather, is this basic property of addition intended to be assumed? Most properties are introduced from first principles, so I don't know if this should be any different. However, directly before the proof Spivak says "It is then possible to find the solution of certain simple equations by a series of steps (each justified by P1, P2, or P3) ...".

Here is the best that I can come up with:

$$x + 3 = 5$$

By P2,

$$x + 3 + 0 = 5 $$

By P3,

$$x + 3 + (-3 + 3) = 5 $$

By P1,

$$x + (3 + (-3)) + 3 = 5$$

Then, (by basic algebra?)

$$x + (3 + (-3)) = 5 + (-3)$$

By P3,

$$x + 0 = 2$$

By P2,

$$x = 2$$