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I was reading my notes and i noticed something a little unusual.

How is $$x + y = y + x$$ not a statement?

The reason that was given in the notes was "we don't know what $x$ and $y$ are, so they are not a statement. In Mathematics, $x$ and $y$ usually represent a real number."

Mathematically, $x + y$ will always be the same as $y + x$ but why are they not considered as a statement?

Statement is a declarative statement that is either true or false but not both

Prajwal Kansakar
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Bryan
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  • What is "Discrete Mathematics" in this context? – Edward Evans Jul 01 '16 at 17:09
  • perhaps a terminology confusion -- $x+y$ and $y+x$ are not statements, but this would be:

    $$x+y = y+x \quad \forall x,y \in \mathbb{R}$$

     – gt6989b Jul 01 '16 at 17:13
  • What do you mean by not both true and false? Is "Fred is a cook" a statement? BTW x + y and y + x will not always be the same if you are not working with real numbers and the additive operation. And even if x + y and y + x will always be the same you do have to state that. – fleablood Jul 01 '16 at 18:16
  • @gt6989b: No, that is not a statement; it is a syntax error. Quantifiers in symbolic logic always come before the formula they range over, so this would be a statement: $$ (\forall x\in \mathbb R)(\forall y\in\mathbb R);x+y=y+x $$ (with possible variations in punctuation, but not with variations in the position of the quantifiers). – hmakholm left over Monica Jul 01 '16 at 19:50

2 Answers2

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The expression $x+y=y+x$ is not a declarative statement that is either true or false. It becomes one (that happens to be true) if you insert specific real numbers for $x$ and $y$ (e.g., $3+\pi=\pi+3$), but as it stands, the symbols $x$ and $y$ do not represent specific real numbers. You might as well write $\square+\triangle=\triangle+\square$, with the understanding that the square and triangular boxes are ‘containers’ waiting to be filled with specific numbers.

You might suppose that it asserts that $x+y$ and $y+x$ are equal no matter what numbers you substitute for $x$ and $y$, but this isn’t how the notation works. If that’s what you want to say, you have to express the no matter what part explicitly:

$$\forall x\,\forall y\,(x+y=y+x)\;,$$

or

for all real numbers $x$ and $y$, $x+y=y+x$.

This potential confusion arises because people, including writers of textbooks, are sometimes sloppy and omit the quantifying expression ($\forall x\,\forall y$ or for all real numbers $x$ and $y$) when they think that it can be reasonably understood from context.

Brian M. Scott
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    In addition to sloppy textbook authors, in many formalizations of first-order logic, the intended semantics of having derived a formula with free variables is that it is true for all value assignments. (But that should probably be considered an internal detail within the formalization). – hmakholm left over Monica Jul 01 '16 at 19:56
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Since it's not stated to apply for all $x$ and $y$, only some unknown particular $x$ and $y$, it declares nothing about the commutativity of addition nor about the properties of your equals sign. And if you know the properties of your equals sign and addition operator then it's not a statement because it declares nothing about $x$ and $y$.

But I think it is a statement under certain circumstances, for example if your addition was not known to be commutative in general then this statement declares that there is some pair $x,y$ such that addition does commute.

The intended statement is probably:

$$x+y=y+x \space\forall (x,y)$$

Which correctly declares that your addition operator is commutative.

it's a hire car baby
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  • what do you mean? it declares that for any value of $x,y$ we must have $x+y=y+x$ – gt6989b Jul 01 '16 at 17:14
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    but it declares that the "+" is always the same, no matter the order of x,y, which is a big statement. imagine $a^b=b^a$ – SAJW Jul 01 '16 at 17:15
  • Hmm... good point! I will fix. – it's a hire car baby Jul 01 '16 at 17:18
  • i dont think you are wrong, as gt6989b said, there must be this "for all foo" in the statement, otherwise it says nothing – SAJW Jul 01 '16 at 17:21
  • Is $x = y + 25$ a statement if x and y have not been introduced? Is the phrase "isoceles triangles are clamptious heffledumps" a statement? I'm not asking rhetorically. I simply don't know, in this context, what the definition of a "statement" is. If x and y haven't been specified to be variables for all real numbers then I think "x + y = y + x" and "x + y = 25" are of equal statement-hood. – fleablood Jul 01 '16 at 17:45
  • @fleablood I think it still is a statement but only in the limited form described in the 2nd paragraph, which is probably not the statement intended. A mathematical statement is defined here http://www.math.ucsd.edu/~benchow/Week1notes.pdf as a sentence that is either true or false but not both. – it's a hire car baby Jul 01 '16 at 17:52
  • But for instance let x be 3 while y be 4, no matter how much it's the same isn't it? It's just not logical. – Bryan Jul 01 '16 at 17:52
  • @TeoChuenWeiBryan are you familiar with abstract algebra? – it's a hire car baby Jul 01 '16 at 17:59
  • @TeoChuenWeiBryan If you take a wall and add red paint and then add blue paint, you have a blue wall. But if you take a wall and add blue paint and then add red paint, you have a red wall. So $a+b$ does not necessarily equal $b+a$. We say this type of addition is not commutative. What you are unaware of is that since you were small you have learnt that conventional addition of real numbers is commutative but this cannot always be taken for granted. When you get on to more advanced maths there are different types of addition, not just the type most people are familiar with. – it's a hire car baby Jul 01 '16 at 18:07
  • If x = 3 and y = 4 then while x + y = 3 + 4 = 7 = 4 + 3 = y + x, that statement doesn't tell us anything about real numbers in general. It just tells us something about the very specific x, y. Not "all x and y". – fleablood Jul 01 '16 at 18:13
  • @fleablood it tells us, that "+" is commutative under the reals – SAJW Jul 01 '16 at 18:53
  • No, it doesn't as it isn't properly formed and does not indicate that x and y represent any real numbers. As it is stated it simply declares that a single thing that is x and single thing that is y happen to be such x + y = y + x but it doesn't say anything about anything other than that one particular x and y. – fleablood Jul 01 '16 at 19:22
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    If I wrote "fred + jam = jam + fred" that statement mean "addition is commutative under the real numbers" or would it mean "fred is a messy eater"? – fleablood Jul 01 '16 at 19:24