Definition:Subtraction

From ProofWiki
Jump to navigation Jump to search

Definition

Natural Numbers

Let $\N$ be the set of natural numbers.

Let $m, n \in \N$ such that $m \le n$.

Let $p \in \N$ such that $n = m + p$.


Then we define the operation subtraction as:

$n - m = p$

The natural number $p$ is known as the difference between $m$ and $n$.


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 difference between the two elements of each tuples 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.


It follows that:

$\forall a, b, c, d \in \N: \eqclass {\tuple {a, b} } \boxminus - \eqclass {\tuple {c, d} } \boxminus = \eqclass {\tuple {a, b} } \boxminus + \tuple {-\eqclass {\tuple {c, d} } \boxminus} = \eqclass {\tuple {a, b} } \boxminus + \eqclass {\tuple {d, c} } \boxminus$


Thus integer subtraction is defined between all pairs of integers, such that:

$\forall x, y \in \Z: x - y = x + \paren {-y}$


Rational Numbers

Let $\struct {\Q, +, \times}$ be the field of rational numbers.


The operation of subtraction is defined on $\Q$ as:

$\forall a, b \in \Q: a - b := a + \paren {-b}$

where $-b$ is the negative of $b$ in $\Q$.


Real Numbers

Let $\struct {\R, +, \times}$ be the field of real numbers.


The operation of subtraction is defined on $\R$ as:

$\forall a, b \in \R: a - b := a + \paren {-b}$

where $-b$ is the negative of $b$ in $\R$.


Complex Numbers

Let $\struct {\C, +, \times}$ be the field of complex numbers.


The operation of subtraction is defined on $\C$ as:

$\forall a, b \in \C: a - b := a + \paren {-b}$

where $-b$ is the negative of $b$ in $\C$.


Extended Real Subtraction

Let $\overline \R$ denote the extended real numbers.

Define extended real subtraction or subtraction on $\overline \R$, denoted $-_{\overline \R}: \overline \R \times \overline \R \to \overline \R$, by:

$\forall x, y \in \R: x -_{\overline \R} y := x -_{\R} y$ where $-_\R$ denotes real subtraction
$\forall x \in \R: x -_{\overline \R} \paren {+\infty} = \paren {-\infty} -_{\overline \R} x := -\infty$
$\forall x \in \R: x -_{\overline \R} \paren {-\infty} = \paren {+\infty} -_{\overline \R} x := +\infty$
$\paren {-\infty} -_{\overline \R} \paren {+\infty} := -\infty$
$\paren {+\infty} -_{\overline \R} \paren {-\infty} := +\infty$

In particular, the expressions:

$\paren {+\infty} -_{\overline \R} \paren {+\infty}$
$\paren {-\infty} -_{\overline \R} \paren {-\infty}$

are considered void and should be avoided.


Ring

Let $\struct {R, +, \circ}$ be a ring.


The operation of subtraction $a - b$ on $R$ is defined as:

$\forall a, b \in R: a - b := a + \paren {-b}$

where $-b$ is the (ring) negative of $b$.


Also known as

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.


In certain historical texts, the term subduction can sometimes be seen.


Also see

  • Results about subtraction can be found here.


Historical Note

The symbol $-$ for subtraction originated in commerce, along with the symbol $+$ for addition, where they were used by German merchants to distinguish underweight and overweight items.

These symbols first appeared in print in $1481$.


Sources