0

I was told that the negation of a negation is not the product of two negations. I understand this to be so because if we prove $-(-x)=x$, we do so using the property of negation: $x+(-x)+(-(-x))=0$, which would equal $x$ and $-(-x)$. I get this part.

However, we know that $-1x=-x$. So if we think of $-(-x)$ as $-1\times(-1x)$, is this not the product of two negations?

Thomas Prévost
  • 579
Amy Lee
  • 3
  • 3
    I would understand "the product of two negations" as $(-x) \cdot (-x)$. Otherwise, this is somewhat ambiguous. – Alexey Uvarov Jan 06 '21 at 15:10
  • I suppose in formal logic, the negation of a truth $p$ is written as $~p$, the negation of this is back to $p$. If one is considering algebraic quantities then the negative of $x$ is written as $-x$ the negative of this is $(-1)(-1)x = (-1)^2 x = x$ –  Jan 06 '21 at 15:14
  • 1
    Use Mathjax. – Wolgwang Jan 06 '21 at 15:19
  • 3
    If $,n(x):= -x,$ then $-(-x) = n(n(x)) = \color{#c00}{n^2}(x),$ is the result of applying the $\rm\color{#c00}{product\ (square)}$ of the negation operation $,n,$ (this product = identity operation, i.e. negation is an involution or reflection). Thus it can be viewed as the result of applying a "product" operator $, n^2 = n\circ n,,$ where multiplication is operator (function) composition in the associated operator algebra. – Bill Dubuque Jan 06 '21 at 15:28

1 Answers1

1

As a matter of fact, yes, $-(-x) = (-1)(-x)$, so in that sense, you could say that $-(-x)$ is the product of two negations.

If you write down the expression $-(-x)$ on a sheet of paper, you haven't written it as a product of two negations, because you haven't written a multiplication sign, nor have you used any other kind of notation that denotes multiplication. But, as you know, negating is the same thing as multiplying by $-1$, so there isn't any important difference between $-(-x)$ (which is written without multiplication) and $(-1)(-x)$ (which is written with multiplication).

In response to:

I was under the assumption that the parentheses after the - denote multiplication, but is this a poor way to think about negation?

In this case, no, the parentheses after the negation sign don't denote multiplication. You can denote multiplication by writing two expressions (that is, two sequences of symbols that represent numbers) next to each other. However, the expression $-(-x)$ does not mean $-$ multiplied by $(-x)$, because $-$ is not an expression (that is, $-$ is not a symbol that represents a number).

That said, I think that no, this is not a poor way to think about negation. Negation is the same thing as multiplying by $-1$ (even if it's not technically defined that way), so, in my opinion, there's nothing wrong with seeing a negation sign and thinking "that means 'multiply by $-1$.'"

Tanner Swett
  • 10,624