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 $p_{1}\wedge p_{2}\wedge\cdots\wedge p_{n}$.
- Please explain what exactly $j=1$ and $p_{j}$ means here.
- What is the name of that notation?
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} $$