Definition:Subtraction
Contents |
Definition
Natural Numbers
Let $\N$ be the set of natural numbers.
Let $m, n \in \N$ such that $m \le n$.
Then we define the symbol "$-$" thus:
- $n - m = p := m + p = n$
This is a translation of the definition Unique Minus in the naturally ordered semigroup.
Integers
The subtraction operation in the domain of integers $\Z$ is written "$-$".
As the set of integers is the Inverse Completion of Natural Numbers, it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the Unique Minus between the two elements of each tuple is the same.
Thus subtraction can be formally defined on $\Z$ as the operation induced on those equivalence classes as specified in the definition of integers.
That is, the integers being defined as all the Unique Minus congruence classes, integer multiplication can be defined directly as the operation induced by natural number multiplication on these congruence classes.
It follows that:
- $\forall a, b, c, d \in \N: \left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus - \left[\!\left[{\left({c, d}\right)}\right]\!\right]_\boxminus = \left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus + \left({- \left[\!\left[{\left({c, d}\right)}\right]\!\right]_\boxminus}\right) = \left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus + \left[\!\left[{\left({d, c}\right)}\right]\!\right]_\boxminus$
Thus integer subtraction is defined between all pairs of integers, such that:
- $\forall x, y \in \Z: x - y = x + \left({-y}\right)$
The General Ring
The proofwiki editor prime.mover has begun this article and plans to complete it some time in the future.
Still to be done: This needs to be expanded to include the definition of the subtraction operation on the general ring. Note the page Definition:Ring Subtraction.
It can be found in prime.mover's Stubs.
If you want to contribute to this page yourself, then please leave a note on prime.mover's talk page informing him of your intentions.
If he does not get back to you within two days, feel free to complete the article yourself.
If you plan to replace a large amount of existing material, then please comment out this material by enclosing it in comment delimiters <!-- --> instead of just deleting it.
Extended Real Numbers
Some notes expressing the care to be taken to avoid $+\infty - (+\infty)$ and note that $-(+\infty)$ is not defined
You can help Proof Wiki by expanding it.
To discuss it in more detail, feel free to use the talk page.
When this page/section has been completed, {{Stub}} should be removed from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Difference
The value $a - b$ (for any of the above definitions) is often called the difference between $a$ and $b$.
In this context, whether $a - b$ or $b - a$ is being referred to is often irrelevant, but it pays to be careful.
Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 2$: Example $2.1$
