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I know that $(a,b)\in R$ means an ordered pair of elements $a$ and $b$ belonging to the set $R$ but sometimes I see some expression like $a R b$ ? What does this notation/expression mean ? How to read $a R b$?

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Some more context would be useful. Symbols seldom have a universal meaning in mathematics. –  Fredrik Meyer Aug 4 '12 at 9:20
    
@FredrikMeyer context is counting theory. –  Geek Aug 4 '12 at 9:23
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$(a,b)\in R$ does not mean that $a$ and $b$ are in $R$, but that the ordered pair of $a$ and $b$ is in $R$. To say that $a$ and $b$ are in $R$, you'd omit the parentheses. –  celtschk Aug 4 '12 at 10:55
    
@celtschk I edited the question. Does it sound right now ? –  Geek Aug 4 '12 at 11:11
    
Yes, now it's correct. –  celtschk Aug 4 '12 at 11:13
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The reason for the notation $aRb$ is that many relations are already written as infix. For example, think of the relation $a<b$ on the set $\{1,2,3,4,5\}$. Actually that relation is given by the set $\{(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)\}$. Now we could give a name to that set, say "$L$" (for "less than") and write "$(a,b)\in L$", but we don't do that. We write "$a<b$". Now think of an arbitrary relation $R$. That arbitrary relation is also given by a set of pairs. But in analogy to relations like $<$ or $\ge$, we want an infix notation, but at the same time want to make clear that it is the relation given by the set $R$. Now one could have selected some general symbol and used that as general "relation" symbol, e.g. $a\rightleftharpoons_Rb$, however at some time someone decided for the minimalist solution of just writing $R$ itself in the middle, i.e. $aRb$, and that caught on. Note however that for specific types of relations (like equivalence relations or orders) usually special symbols ($\sim$, $\preceq$) are used, possibly with index $R$.

So in short, $aRb$ is exactly equivalent to $(a,b)\in R$, however emphasizes that this is a relation.

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and how it it read as ? "Some relation R exits between a and b?" –  Geek Aug 4 '12 at 11:44
    
@Geek: I'd just read the letters in sequence, but I guess that's not the standard way to read it. –  celtschk Aug 4 '12 at 12:06
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R usually denotes some relation, so it means that $a$ and $b$ are associated under relation R. If relation is represented by some subset of $C \subset A \times B$, then $a R b$ means that $(a,b) \in C$.

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What does (a,b)∈C exactly mean ? Does it simply mean a and b belong to set C or something different ? –  Geek Aug 4 '12 at 9:09
    
@Geek: No, $(a,b)$ represents an ordered pair, so $(a,b) \neq (b,a)$ unless $b=a$ and $(x_1, x_2) = (y_1, y_2)$ if and only if $x_1 = y_1$ and $x_2 = y_2$. –  user5501 Aug 4 '12 at 9:10
    
What does ordered pair mean in context of the set {a,b,c} does it mean a<b<c ? –  Geek Aug 4 '12 at 9:25
    
@Geek: Well, without some more context, I'd say that ordered pair in context of that set is just an ordered pair of elements of that set, i.e. $(x,y)$ where $x\in \{ a,b,c \}$ and $y\in \{a,b,c\}$ or said differently just the set $\{a,b,c\} \times \{a,b,c\}$. There exist order relations, but they're not strictly related to ordered pairs. –  user5501 Aug 4 '12 at 9:31
    
Please see my edited question. I wanted to know how to read the expression $a R b$ –  Geek Aug 4 '12 at 11:09
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A relation $R$ on a set $S$ is just a subset of $S\times S$. and we say $a$ R $b$ in other words a and b are R related. for example I give a relation R on $\mathbb{Z}$ such that $aRb\Leftrightarrow 5|(a-b)$

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That relation partitions integers into classes modulo $5$, though it is an example of a particularly nice-behaving relation, i.e. it's an equivalence relation. –  user5501 Aug 4 '12 at 9:16
    
yes, this is an equivalence relation, reflexive, symmetric and transitive –  miosaki Aug 4 '12 at 9:17
    
@KutukKatuk What does symbol between 5 and a-b mean ? How it is called ? –  Geek Aug 4 '12 at 11:16
    
5 $|$ (=divides) (a-b) –  miosaki Aug 4 '12 at 12:01
    
@Geek: That symbol means "divides" (as in "5 divides a minus b"), and it says that $(a-b)$ is a multiple of $5$. –  celtschk Aug 4 '12 at 12:08
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Like the above poster said, $a R b$ usually means some kind of relation.

Another notation is $a \sim b$, which is frequently used to denote an equivalence relation.

A concrete example is $aRb$ if and only if $a-b$ is even.

Note that this is an equivalence relation:

aRa since $a-a=0$ is even. (reflexive)

aRb implies bRa since $a-b$ even implies $b-a$ even. (symmetric)

aRb, bRc implies aRc since $a-b$ even and $b-c$ even implies $a-c$ even. (transitive)

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$aRb$ means the same as $(a,b) \in R$

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