what does this notation mean 
If $p_1, p_2, ⋯, p_n$ are $n$ propositions, explain why
$$\bigwedge_{i=1}^{n-1}\bigwedge_{j=i+1}^n(\lnot{p_i}\lor\lnot{p_j})$$
is true iff at most one of $p_1, p_2, ⋯, p_n$ is true.

My questions are:

*

*the two continous $\wedge$ symbols looks strange,what does it mean

*what's the expansion of this notation

 A: This is just like sigma notation. It means conjunction.
Like $$\sum_{i=0}^n a_i = a_0 + a_1 + a_2 + \dots + a_n$$ here we don't have addition but conjunction (and)
$$\bigwedge_{i=1}^{n} P_i = P_1 \wedge P_2 \wedge P_3 \wedge \dots \wedge P_n$$
For sums and products, the conventional notations Sigma ($\Sigma$) and Pi ($\Pi$) are used. For other associative operations, we just use upscaled versions of the operator.
Other such examples are: disjunction $\bigvee$, union $\bigcup$, intersection $\bigcap$, disjoint union $\bigsqcup$.
A: The first question had been well explained by @Stefan Octavian, for your second question:



*what's the expansion of this notation?


\begin{align}
&\bigwedge_{i=1}^{n-1}\bigwedge_{j=i+1}^n(\lnot{p_i}\lor\lnot{p_j})\\
\equiv&\bigwedge_{j=2}^n(\lnot{p_1}\lor\lnot{p_j})\land\dots\land\bigwedge_{j=n}^n(\lnot{p_{n-1}}\lor\lnot{p_j})\\
\equiv&\hspace{2.5ex}(\lnot{p_1}\lor\lnot{p_2})\land\dots\land(\lnot{p_1}\lor\lnot{p_n})\\
&\land(\lnot{p_2}\lor\lnot{p_3})\land\dots\land(\lnot{p_2}\lor\lnot{p_n})\\
&\hspace{8.5ex}\vdots\\
&\land(\lnot{p_{n-1}}\lor\lnot{p_n})
\end{align}
Now why this is true if and only if that at most one of $p_1,p_2,\dots,p_n$ is true?
Hint:
Suppose for $i\neq j$ in $\{1,2,\dots,n\}$ we have that both $p_i,p_j$ is true, then $(\lnot p_i\lor \lnot p_j)$ is false, and we know that any clause of a conjunction is false then the whole conjunction is false.
Similarly if the whole conjunction is false, then at least one clause must be false,   but for any clause $(\lnot p_i\lor \lnot p_j)$ to be false, it follows that $p_i,p_j$ are both true.
This would be the general idea of your proof.
