Name of the arithmetic property if a=b then a+c = b+c?

Question

Properties of arithmetic operations such multiplication and division have names. For example:

$a + b = b + a$ (commutativity)

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

and so on

is there a name for

if $a=b$ then $a+c = b+c$ ?

Closely related Does "Doing a thing to both sides of an equation" have a name? See esp. my comments on the (misleading) accepted answer there. — Bill Dubuque, May 16 '19 at 17:41

score 3 · Answer 1 · answered May 16 '19 at 08:30

3

I don't think it has a name. It's just a special case of $a=b\implies f(a)=f(b)$.

answered May 16 '19 at 08:30

J.G.

115,835

score 1 · Answer 2 · answered May 16 '19 at 08:33

1

Adding $c$ to both sides of an equation is a so-called "equivalence transformation" ; that is a transformation that does not change the set of solutions of the equation.

answered May 16 '19 at 08:33

Peter

84,454

score 0 · Answer 3 · answered May 16 '19 at 08:31

0

I don't think it has a name. The converse rule, however: $$ a + c = b + c\implies a = b $$ is called cancellation. You could argue that your rule is in some sense cancellation for subtraction, but it takes more work than it's worth, in my opinion.

answered May 16 '19 at 08:31

Arthur

199,419

score 0 · Answer 4 · answered May 16 '19 at 08:53

Well, the generalization would be monotonicity of addition:

$$a\leq b\Rightarrow a+c\leq b+c.$$

Note that if $a\leq b\wedge b\leq a\Rightarrow a=b$ and

$a\leq b :\Leftrightarrow \exists c: a+c=b.$

Here if you take as underlying set the ring of integers, the field of rational, real or complex numbers, you are fine.

score 0 · Answer 5 · answered May 16 '19 at 09:38

The book Introduction to Mathematical Logic by Elliot Mendelson, which was the textbook for my first course in mathematical logic, has a section on "First-Order Theories with Equality" starting on page 75. The book has two requirements for a theory with equality. The second requirement is that this is a theorem:

$$x=y \supset (\mathscr{A}(x,x) \supset \mathscr{A}(x,x))$$

where $x$ and $y$ are any variables, $\mathscr{A}(x,x)$ is any well-formed formula, and $\mathscr{A}(x,y)$ arises from $\mathscr{A}(x,x)$ by replacing some, but not necessarily all, free occurences of $x$ by $y$, with the proviso that $y$ is free for the occurences of $x$ which it replaces. (And of course, the "$\supset$" symbol means implication.) The name of this axiom schema is

Substitutivity of Equality

Your property is a particular theorem included in this schema. The schema is a generalization of @J.G.'s answer, but it has a name.

score 0 · Answer 6 · 2019-05-16T19:10:39.723

See : Transforming equations : rules governing the use of '<=>' and '=>' . and Mauro Allegranza's answer.

What follows is simply a development of J.G 's answer.

Define a function f , say from the set of real numbers to the set of real numbers , such that f(x) = x+2; that is, in terms of relation, f is the relation such that :

f = {(x,y) | y = x + 2 }.

A relation R is a function iff :

no two different ordered pairs belonging to R have the same first element.

In other words, a relation R is a function iff :

all orders pairs belonging to R that have the same first element also have the same second element.

Now, the ordered pairs ( a, z) and (b, z') such that z = a+2 and z'= b+2 belong to the function f .

These two pairs have the same first element, since, by hypothesis: a = b.

Consequently, they also have the same second element, which means that z = z'.

According to the definitions of z and of z', it means that :

                             a+2 = b+2

Name of the arithmetic property if a=b then a+c = b+c?

6 Answers6