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.

  • $\begingroup$ For $n=3$, does the last term of your formula mean either $a_2 \land a_3$ or $a_2 \lor a_3$? $\endgroup$ – Doyun Nam Jan 31 '19 at 2:38
  • $\begingroup$ it was a typo i fixed it $\endgroup$ – user614735 Jan 31 '19 at 2:38
  • $\begingroup$ This is perhaps not the most elegant, but my first instinct was to use $\bigvee\limits_{\{i,j\}\in\binom{[n]}{2}}(a_i\wedge a_j)$, using the notation that $[n]=\{1,2,3,\dots,n\}$ and $\binom{A}{k}$ with $A$ a set is the set of subsets of size $k$ of $A$. This doesn't work for $n=1$, but should work for all larger $n$. To be fair, the meaning I use for the notation $\binom{A}{k}$ is not widely used outside of smaller circles in combinatorics. $\endgroup$ – JMoravitz Jan 31 '19 at 2:40

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.


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