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Is there a name for this simple operation/function?

$$(a, b) \mapsto a + (1 - a)b$$

The operator takes two real numbers that are between $0$ and $1$, and returns a real number per the formula. Intuitively, it moves the $a$ value towards $1$ by using $b$ as the fraction of the remaining distance to travel. Note that the order of the operands doesn't matter, since $a + (1 - a)b$ rearranges to $b + (1 - b)a$. Another perspective is that it's the sum, minus the product:

$$(a, b) \mapsto a + b - ab$$

So, for example:

  • $(0.5, 0.5) \mapsto 0.75$
  • $(0.3, 1) \mapsto 1$
  • $(0.47, 0.61) \mapsto 0.7933$

Thanks!

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    I haven't seen it, but I would interpret it, instead of moving $a$ as moving $b$. This function returns the convex combination of $1$ and $b$ with weights $a$ (and $1-a$). – orole Jan 15 '18 at 13:57
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    The function $f(x,y,t) = (1-t)x+ty$ is usually called linear interpolation, also mix or lerp sometimes. It returns $x$ if $t=0$, $y$ if $t=1$, and some convex combination of $x$ and $y$ if $t\in (0,1)$. You have a special case of $f(b,1,a)$. – lisyarus Jan 15 '18 at 14:09
  • $1-(1-a)(1-b)$, does that help? – Dominique Jul 17 '23 at 13:31

1 Answers1

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Given the unnamed involution $\,r(x) := 1-x,\,$ then the function (binary operation) $$ f(a, b) := a + (1 - a)b = a+b-ab $$ is the conjugation of the multiplication operation by the involution $\,r.\,$ That is, $$ r(f(a,b)) = 1-a-b+ab = (1-a)(1-b) = r(a)r(b). $$ This operation is often used to give an example of a commutative associative operation that is not either addition or multiplication. For example MSE question 526300, MSE question 721323. Similarly the related operation $\,a+b+ab\,$ in MSE question 1509381 where it is denoted by $\,a\odot b\,$ but not otherwise named is much more common.

EDIT (23 Jul 2023): George Boole originally invented his eponymous algebra in the 19th century using variables taking on only values of $0$ or $1$. In his context, they represents logical False and True, respectively, the operation $\,r\,$ represents logical negation, while multiplication represents logical conjunction. The remaining operator $\,f\,$ represents logical disjunction. You can think of $\,f\,$ defined on the interval $\,[0,1]\,$ as generalizing this in the context of fuzzy logic.

The answer to your question

Is there a name for this simple operation/function?

is that it is the logical disjunction operation in fuzzy logic.

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