Q:

Sum and product notation

I'm working with logic, but I need help with notation.
I'll give examples of what I want, because you will see the pattern. For each $n$, I want to perform "AND" on each pair, and OR all of the pairs together.
For $n = 1$, I want $a_{1}$
For $n = 2$, I want $(a_{1} \wedge a_{2})$
For $n = 3$, I want $(a_{1} \wedge a_{2}) \vee (a_{1} \wedge a_{3}) \vee (a_{2} \wedge a_{3})$
I want to write this with formal notation. I tried 
$$\bigvee_{i=1}^{n-1} (a_{i} \wedge a_{i+1}), $$
but it doesn't work for $n = 3$. Any ideas? I think it might involve two AND/OR's, and I suspect that the second AND/OR will begin at the outside AND/OR's index. 

A:

For $n \geq 2$, 
$$\bigvee\limits_{j=2}^n \big(\bigvee\limits_{i=1}^{j-1}(a_i \land a_j) \big)$$
might satisfy your formula. 

