Word for equivalence preserving transformations of equations

I am searching for a mathematical term describing an algebraic manipulation of an equation which preserves equivalence. So while adding 2 to both sides of an equation results in an equivalent equation, squaring both sides does not.

In German, there is the word "Äquivalenzumformung" (literally: equivalence transformation). Is there a similar word in English?

• You will have better luck on Mathematics Stack Exchange (math.stackexchange.com) but you will find you have to define 'equivalence' better. – Spencer Jun 14 '16 at 23:14
• Uh, hmm, isomorphic transformations? Or algebraic transformations? – Dan Bron Jun 15 '16 at 2:56
• @Spencer Thanks, I'll try. ('equivalence' is a perfectly unambiguous mathematical term though!) – 黄雨伞 Jun 16 '16 at 7:07
• Is this category theory? Then it's just 'isomorphism' or 'bijection'. Also, this is highly technical vocabulary and even if math.SE sent you back here, really, they are the better ones to know. – Mitch Aug 18 '16 at 2:58
• When you use matrices to represent and solve systems of linear equations, "elementary row operations" may be performed on a matrix to solve the system of linear equations. – Graffito Sep 18 '16 at 19:07

Though I agree this question will likely find its answer more easily on Mathematics SE, I'll try to answer briefly here, and forewarn you of the rather picky tendencies of mathematical language.

The best I can offer is a Homomorphism (Wikipedia), which is a term from abstract algebra, but it sounds as thought it might fit your criteria.

PS: You assert in a comment that "equivalence" is a well-defined mathematical concept, but that is only true in-context. See the Mathematics section of Wikipedia's Equivalence - Disambiguation, for example. Equivalence can mean much more than whether two sides of an equation produce the same result.

• Thank you. I tried it on Mathematics SE, and they said there probably were no such word. (By the way, I am speaking of logical equivalence, the relation that is commonly denoted by "<==>".) – 黄雨伞 Jul 16 '16 at 8:00
• @黄雨伞 Can you give a link to the math.se question? – Mitch Aug 18 '16 at 2:57

You may be describing symmetry https://en.wikipedia.org/wiki/Symmetry_in_mathematics though it sounds like "Äquivalenzumformung" may be more specific.

• No, this is something else. Talking about functions, I'm seeking for a term describing functions \$f\$ for which \$x = y \iff f(x) = f(y)\$, but that \$f\$ is symmetric means that \$S(f(x)) = S(x)\$ for some kind of structure S; 'symmetry' is a term with uses in many fields. And yes, talking about functions, there is the term 'bijective', so instead of 'making a Äquivalenzumformung' we could say 'applying a bijective function to both sides of the equation' but this is confusing and furthermore has no equivalent if we weren't dealing with a equation, but, say, with an inequation. – 黄雨伞 Jun 21 '16 at 11:13
• @黄雨伞 You should make that part of your question rather than a throw away comment. – Mitch Aug 18 '16 at 2:56

I am not aware of a similar word in English, but I believe the terms you are looking for are as follows:

Addition Property of Equality: If a, b, and c are numbers and a = b, then a + c = b + c.

Multiplication Property of Equality: If a, b, and c are numbers and a = b, then ac = bc.

A few technicalities aside, in general you can apply the same function f(x) to both sides of any equation: if a = b, then f(a) = f(b).

So maybe you are looking for Function Property of Equality. Unfortunately not one word. NOTE: https://en.wikipedia.org/wiki/Equality_(mathematics) refers to this as the Substitution Property of Equality.

By the way, you can square both sides of an equation while maintaining equality: if a = b, then a2 = b2.

• But if \$a^2 = b^2\$, \$a\$ may not equal \$b\$ (one could be negative. – Mitch Aug 18 '16 at 14:21
• True. That's one of the technicalities to which I was referring. As I'm sure you know, the fix is to use the function "absolute value of the square root". Anyway, "function property of equality" or "substitution property of equality" is the best answer I've been able to come up with at the relevant level of mathematics. It's a bit surprising there doesn't seem to be a single word for it. – Richard Kayser Aug 18 '16 at 14:36

Here are some sentences that include your idea:

We can rewrite Eq. (1) as Eq. (2).

Eq. (1) is equivalent to Eq. (2).

Such-and-so (e.g. substituting u = 1/v) gives us the equivalent formulation ....

If you're trying to make a particular sort of sentence, please give us the sentence you're aiming for, with a blank in the place where the word or expression would go.

• An example sentence would be "If we square both sides of the equation we might lose one or more solutions, because squaring is not a …" – 黄雨伞 Sep 1 '16 at 16:08