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STATEMENT:

$$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$

As per my knowledge, a sentence is said to be a mathematical statement iff it fulfills both of the following criteria:

  1. The sentence should be declarative.

  2. It should have a definite truth value.

The given statement seems to fullfill both of the above criteria - it is indeed declarative in nature and is true for all values of a and b (both real and complex). It should, hence, be a mathematical statement but the textbook says otherwise. Which part am I missing out on?

Source: Mathematical Reasoning - Writing and Proof (Version 2.1) by Ted Sundstrom, Progress Check 1.1 - Question 8

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CalIMpire257
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    It does not say what $a,b,c$ are. – geetha290krm Sep 30 '22 at 05:57
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    I'm just spitballing here, but maybe the problem is that the statement doesn't clarify what objects we're dealing with. Is this an equality in the real numbers, the complex numbers, polynomials in the variables $a$ and $b$, or something else? – Christian E. Ramirez Sep 30 '22 at 05:59
  • Hey, but addition and multiplication operations can only be performed over "numbers", isn't it? Since they've already performed those operations over a and b, shouldn't it imply that a and b are numbers? – CalIMpire257 Sep 30 '22 at 05:59
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    It is an open formula. – Damian Sep 30 '22 at 06:00
  • @C-RAM isn't it true irrespective of which type of equality it represents? – CalIMpire257 Sep 30 '22 at 06:04
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    @CalIMpire257 Addition and multiplication can be performed on any field, (and can even be expanded to any vector space or other algebraic structures, or could just be defined completely abstractly without respect for such notions), and in such structures the statement might not hold, even though it does on $\mathbb R$. – Logan M Sep 30 '22 at 06:04
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    It's missing a quantifier, such as "For all $a$ and $b \in \mathbb{R}$." The same problem would apply to a sentence such as "$x = x$." As @Damian points out, the formula is open. – Brian Tung Sep 30 '22 at 06:05
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    The statement may be false when $a,b,c$ are matrices – Hagen von Eitzen Sep 30 '22 at 06:05
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    “shouldn’t it imply that $a$ and $b$ are numbers?” Who knows, they could be matrices and then the formula might be false. Or they could be something else entirely that I’m not even aware of, just as some people aren’t aware of matrices. Without saying what $a$ and $b$ are, there’s no way to know what this statement means without being a mind reader. – littleO Sep 30 '22 at 06:06
  • I get it now, Thank you so much for helping out! (to everyone involved in this comment section) – CalIMpire257 Sep 30 '22 at 06:07

1 Answers1

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It is not a mathematical statement because its truth value is unclear. It is not known what $a$ and $b$ are. For example, if $a$ and $b$ are real numbers, then sure, the equation is correct. But if they are elements of a non-commutative structure, then the equation may not hold anymore.

In other words, the statement

$\forall a,b\in\mathbb R: (a+b)^3=a^3+3a^2b+3ab^2+b^3$

is true, but the statement

$\forall a,b\in\mathbb R^{2\times 2}: (a+b)^3=a^3+3a^2b+3ab^2+b^3$

is not true. For example, you could take $$a=\begin{bmatrix}0&1\\0&0\end{bmatrix}, b=\begin{bmatrix}0&0\\1&0\end{bmatrix}$$

for a simple counterexample, since the left hand side evaluates to $a+b$ while the right hand side evaluates to a zero matrix.

5xum
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