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I am currently reading "Discrete Mathematics and Its Applications, 7th ed", p.28.

From there I read:

We will sometimes use the notation $\bigvee_{j=1}^{n} p_{j}$ for $p_{1}\vee p_{2}\vee\cdots\vee p_{n}$ and $\bigwedge_{j=1}^{n} p_{j}$ for $p_{1}\wedge p_{2}\wedge\cdots\wedge p_{n}$.

  1. Please explain what exactly $j=1$ and $p_{j}$ means here.
  2. What is the name of that notation?
vasili111
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  • Its seems like we talk about boolean functions here (I think?). If yes, p consists of an array of "n" values denoted "p_j". The first part will mean p_1 OR p_2 OR p_3 ... OR p_n while the second part means p_1 AND p_2 AND p_3 .... AND p_n. Not sure which part of the notation you refer to though ;/ – Snifkes Aug 21 '18 at 15:35
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    $j$ is the dummy index. $\vee_{j=1}^n p_j$ means that $j$ runs through the numbers $1,2,\ldots,n$ (1 is the smallest value of $j$, $n$ is the largest), so it's just shorthand for $p_1 \vee p_2 \vee p_3 \vee \ldots \vee p_n$. It's like summation notation. – A. Goodier Aug 21 '18 at 15:37

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As others have mentioned, the $j$ is just some dummy variable that runs through a particular index. I'm not sure if there's a common name for this notation, it's usually just called indexing or index notation.

As rbird mentions, this is very similar to the summation notation. To remind you, this is $$ \sum_{j = 1}^{n} x_{j} = x_{1} + x_{2} + \cdots + x_{n}. $$ The $j = 1$ on the bottom indicates that the summation starts at the $x_{j} = x_{1}$ term, and goes through each of the integers up to and including $n$. If you have a set of elements labelled $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$, then the notation $x_{i}$ or $x_{j}$ is commonly used to refer to an arbitrary element without specifying which.

Note that $j$ can be replaced with any other dummy variable and that if we change $j = 1$ to $j = 2$ (for example), this indicates that we start the summation/index at $j = 2$ instead of $j = 1$. An explicit example is $$ \bigvee_{i = 7}^{9} p_{i} = p_{7}\vee p_{8}\vee p_{9} $$ which is exactly the same as $$ \bigvee_{\ell = 7}^{9} p_{\ell} \qquad\text{and}\qquad \bigvee_{\gamma = 7}^{9} p_{\gamma}. $$ The choice of which dummy letter to use ($i, \ell, \gamma$) is up to you, though it is common to see $i, j, k$ used for indexes.

The notation $$ {\textstyle \bigvee_{i = 7}^{9} p_{i}} \qquad\text{and}\qquad \bigvee_{i = 7}^{9} p_{i} $$ are both correct: the left is text style which is common seen written inside a body of text, while the right is display style which is common seen written on its own line. They are equivalent.

You should note that \bigvee_{i = 7}^{9} p_{i} gives the left one when you use it between single $'s (inline maths), and that \bigvee_{i = 7}^{9} p_{i} gives the right one when you use it between double $$'s:

$$ \begin{array}{cc} \texttt{\$\bigvee_{i = 7}^{9} p_{i}\$} & {\textstyle \bigvee_{i = 7}^{9} p_{i}}\\ & \\ \texttt{\$\$\bigvee_{i = 7}^{9} p_{i}\$\$} & {\displaystyle \bigvee_{i = 7}^{9} p_{i}} \end{array} $$

Bilbottom
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  • The proposition you wrote bove and $\bigvee_{i=7}^{9} p_{i}$ means exactly the same, right? I mean I can place 9 and i=7 above/below or right of $\bigvee$, both are the same, right? – vasili111 Aug 21 '18 at 15:56
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    @vasili111 Absolutely, they're both a short way of writing $p_{7} \vee p_{8} \vee p_{9}$. Note that there is nothing special about this example, it is completely arbitrary. – Bilbottom Aug 21 '18 at 15:58
  • I mean I can place 9 and i=7 above/below or at right top/bottom of $\bigvee$, both are the same, right? – vasili111 Aug 21 '18 at 16:00
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    @vasili111 Yes, that's correct -- they correspond to inline maths and display maths rendering. See my update. – Bilbottom Aug 21 '18 at 16:04
  • Great explanation. Thank you. – vasili111 Aug 21 '18 at 17:34