How two interpret two indices on a wedge sign?

Question

$$\psi_n:=\bigwedge_{1\leq t\leq n}\left(\bigvee_{1\leq i<j\leq n+1}P_{i,t}\vee P_{j,t}\right)$$ I have trouble understanding the second wedge part of this formula because of its two indices. What would it be written out for something like $\psi_{2}$ ?

Answer 1

This is analogous to summation notation $\sum_{1\le i\le n}$ or similar.

For example, $$\bigwedge_{1\le i\le 3}\theta_i$$ is shorthand for $$\theta_1\wedge\theta_2\wedge\theta_3.$$ The "big wedge" (or "big vee") notation really shines when it's iterated, as it is here; for example, $$\bigvee_{1\le i\le 3}\bigwedge_{1\le j\le 3}\theta_{i,j}$$ is a very compact way of writing $$(\theta_{1,1}\wedge\theta_{1,2}\wedge\theta_{1,3})\vee(\theta_{2,1}\wedge\theta_{2,2}\wedge\theta_{2,3})\vee(\theta_{3,1}\wedge\theta_{3,2}\wedge\theta_{3,3}).$$

And, as with summation notation, "inner" indices can depend on "outer" variables - think about something like $\sum_{1\le i\le 3}\sum_{1\le j\le i}$. Note that this last subtlety is not present in the example you give, but it is worth mentioning.

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As to the specific example you consider, we have $$\psi_2\equiv \bigwedge_{1\le t\le 2}(\bigvee_{1\le i< j\le 3}(P_{i,t}\vee P_{j,t}))$$

$$\equiv(\bigvee_{1\le i< j\le 3}(P_{i,1}\vee P_{j,1}))\wedge(\bigvee_{1\le i< j\le 3}(P_{i,2}\vee P_{j,2})).$$

And this in turn unpacks to $$[(P_{1,1}\vee P_{2,1})\vee(P_{1,1}\vee P_{3,1})\vee(P_{2,1}\vee P_{3,1})]\quad\wedge \quad[(P_{1,2}\vee P_{2,2})\vee(P_{1,2}\vee P_{3,2})\vee(P_{2,2}\vee P_{3,2})]$$ (spaces added for ease of reading). Note how "$\bigvee_{1\le i<j\le 3}$", just like "$\sum_{1\le i<j\le 3}$," is used to capture a whole collection of pairs of elements; the key point is that there are exactly three pairs $(i,j)$ satisfying $1\le i<j\le 3$, namely $(1,2),(1,3),(2,3)$.

Answer 2

The notation "$\>1\leq i<j\leq n+1\>$" under the big vee (not "wedge") $\bigvee$ defines the index set over which we have to sum. This set is not of the usual form $[n]:=\{1,2,3,\ldots, n\}$, or similar, but is the set of all pairs $(i,j)$ of numbers $i<j$ taken from $[n+1]$. For each such pair $(i,j)$ we have to form $P_{i,t}\vee P_{j,t}$.

When $n=2$ we have $$\psi_2=\left(\bigvee_{1\leq i<j\leq3}P_{i,1}\vee P_{j,1}\right)\>\wedge\>\left(\bigvee_{1\leq i<j\leq3}P_{i,2}\vee P_{j,2}\right)\ .$$ Furthermore $$\bigvee_{1\leq i<j\leq3}P_{i,1}\vee P_{j,1}=(P_{1,1}\vee P_{2,1})\vee(P_{1,1}\vee P_{3,1})\vee(P_{2,1}\vee P_{3,1})\ .$$