I'm reading Kenneth H. Rosen Book on Discrete Math, there is a question in the first section of the first chapter. There is question, which has some notation like this.
$$\bigwedge\limits_{i=1}^{n-1}\bigwedge\limits_{j=i+1}^{n}(\lnot p_i \lor \lnot p_j)$$
Can anyone give me little explanation what's meaning of it.
I have no idea what it is trying to say.
I don't really know the context of the statement you provided, but perhaps the best way to understand what's going on is to 'expand' the notation out. As such, we want to consider the expression as $$ \bigwedge_{i=1}^{n-1}\bigwedge_{j=i+1}^{n}(\lnot p_i \lor \lnot p_j) = \bigwedge_{i=1}^{n-1}\left(\bigwedge_{j=i+1}^{n}(\lnot p_i \lor \lnot p_j)\right). $$ So, taking the notation at its word, expand the big bracket as $$ \bigwedge_{i=1}^{n-1}\left(\bigwedge_{j=i+1}^{n}(\lnot p_i \lor \lnot p_j)\right) = \bigwedge_{i=1}^{n-1}\bigg( (\lnot p_i \lor \lnot p_{i+1})\land (\lnot p_i \lor \lnot p_{i+2}) \land \cdots \land (\lnot p_i \lor \lnot p_{n-1}) \land (\lnot p_i \lor \lnot p_{n}) \bigg). $$ Then, if $n=10$ (why not!) we have the behemoth of an expression given as \begin{align*} \bigg [(\lnot p_1 \lor \lnot p_{2})\land (\lnot p_1 \lor \lnot p_{3}) \land \cdots \land (\lnot p_1 \lor \lnot p_{8}) \land (\lnot p_1 \lor \lnot p_{10}) \bigg ] &\land \\ \bigg [\hphantom{(\lnot p_2 \lor \lnot p_{3})}(\lnot p_2 \lor \lnot p_{3}) \land \cdots \land (\lnot p_2 \lor \lnot p_{8}) \land (\lnot p_2 \lor \lnot p_{10}) \bigg ] &\land \\ &\vdots \\ \bigg [(\lnot p_8 \land \lnot p_9) \land (\lnot p_8 \land \lnot p_{10})\bigg] &\land \\ \bigg [(\lnot p_9 \land \lnot p_{10})\bigg ].\end{align*} As you can see, we are computing all possible (not $p_i$ or not $p_j$ expressions) and 'anding' them all together. The notation is set out in such a way as to not get repeated expressions such as $(\lnot p_1 \lor \lnot p_{2})$ and $(\lnot p_2 \lor \lnot p_{1})$, which are, of course, the same thing and hence superfluous.