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Let's say we have the following definition:

A relation $\rho$ on a set $A$ is called transitive if

$a \rho b \wedge b \rho c \Rightarrow a \rho c$

is true for all $a,b,c \in A$.

Another example: $\forall a, b, c \in A (\dots)$.

Do these statements automatically mean/give the condition that $a \neq b \neq c$ or would that have to be specified separately? I know this question is very basic, but I did not find an answer when searching in my words, so I thought I'd ask here and maybe the answer could help others along the way.

Thanks a lot!

Joe
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Aaron
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    A statement of the form $\forall a, b, c \in A(P(a,b,c))$ means that for every choice of $a$, $b$, and $c$ in $A$, the predicate $P(a,b,c)$ holds. It is not assumed that $a\neq b \neq c$. – Joe Jul 12 '23 at 15:34
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    Nor, indeed, is it assumed that $a\ne c$. :P – Xander Henderson Jul 12 '23 at 15:35
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    (I'll note that my comment, while facetious, kind of gives an example of why we care. The relation $\ne$ is not transitive. A simple example which demonstrates this is to observe that $1 \ne 2$ and $2 \ne 1$, but it is not true that $1\ne 1$.) – Xander Henderson Jul 12 '23 at 15:38
  • To give an example from what you appear to be studying, the universal relation $\rho$ on $A$ is the relation which satisfies $\forall a,b\in A(a\rho b)$. In particular, it satisfies $\forall a\in A(a\rho a)$. – Joe Jul 12 '23 at 15:38
  • You can apply instanciation successively for each variable quantified with a $\forall$, replacing them with the same element. – NDB Jul 12 '23 at 15:40
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    How to express “There exist $a,b,$ and $c$, all different, such that…” is actually a frequently-asked question here. For example: 1 2 3 – MJD Jul 12 '23 at 19:32

1 Answers1

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No, writing $\forall a,b,c\in A$ does not imply $a,b,c$ are different. Nor we would want it to mean that. For example, let's say we want to write axiom of distributivity: $$(\forall a,b,c \in A)\ a(b+c) = ab + ac.$$ When we write it down, we'd also like to include cases like $ c= a$, where it would read $a(b+a) = ab + a^2$, but if we assumed that all $a,b,c$ are different, we would have to list this separately, which would be tedious.

Instead, when we don't want to include cases where some of the variables are equal, we write that explicitly. Let's say that we have a set $\{x_1,x_2,\ldots,x_n\}$ and we want to write down that all $x_i$'s different. We could write it like: $$(\forall i,j\in\{1,2,\ldots,n\})\ x_i\neq x_j,\, i\neq j.$$ Note that we have to write the condition $i\neq j$ because $x_i = x_i$ for all $i$.

Ennar
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    And this is generally true in mathematics. If we use two variables $x,y$, although the labels are different, it doesn't exclude the case where the values of $x$ and $y$ are the same. For example when you want to solve linear system $x + y= 2$, $x+2y = 3$. There is a unique solution $x = y = 1$. – Ennar Jul 12 '23 at 16:02