Propositional logic notation conversions and naming.

Question

Question 1

For example for: $$ \bigvee_{i = 7}^{9} p_{i} $$

we write: $$ \bigvee_{i = 7}^{9} p_{i} = p_{7}\vee p_{8}\vee p_{9} $$

What we write for? $$\bigwedge_{i=1}^{9} \bigwedge_{n=1}^{9} \bigvee_{j=1}^{9}~p(i,j,n)$$

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Question 2

What is called that form?

$$ \bigvee_{i = 7}^{9} p_{i} $$

And what is called that form?

$$ p_{7}\vee p_{8}\vee p_{9} $$

Answer 1

\begin{equation} \bigwedge_{i=1}^{2} \bigwedge_{n=1}^{2} \bigvee_{j=1}^{2}~p(i,j,n)=\\ =\left(\bigwedge_{n=1}^{2} \bigvee_{j=1}^{2}\ p(1,j,n)\right) \wedge \left(\bigwedge_{n=1}^{2} \bigvee_{j=1}^{2}\ p(2,j,n)\right)=\\ =\left[\left(\bigvee_{j=1}^{2}\ p(1,j,1)\wedge \bigvee_{j=1}^{2}\ p(1,j,2)\right)\right] \wedge \left[\left( \bigvee_{j=1}^{2}\ p(2,j,1)\wedge \bigvee_{j=1}^{2}\ p(2,j,2)\right)\right]=\\ =(p(1,1,1)\vee p(1,2,1))\wedge (p(1,1,2)\vee p(1,2,2))\wedge (p(2,1,1)\vee p(2,2,1))\wedge (p(2,1,2)\vee p(2,2,2)) \end{equation}

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This is the compact form: $$\bigvee_{i=7}^{9}p_i$$

This is the expanded form: $$p_7\vee p_8\vee p_9$$

Answer 2

Answer to question 1:

\begin{align} \bigwedge_{i=1}^{9} \bigwedge_{n=1}^{9} \bigvee_{j=1}^{9}~p_{i,j,n} &= \bigwedge_{i=1}^{9} \bigwedge_{n=1}^{9} (p_{i,1,n} \lor p_{i,2,n} \lor p_{i,3,n} \lor p_{i,4,n} \lor p_{i,5,n} \lor p_{i,6,n} \lor p_{i,7,n} \lor p_{i,8,n} \lor p_{i,9,n} ) \\ = \bigwedge_{i=1}^{9} &\big(( p_{i,1,1} \lor p_{i,2,1} \lor p_{i,3,1} \lor p_{i,4,1} \lor p_{i,5,1} \lor p_{i,6,1} \lor p_{i,7,1} \lor p_{i,8,1} \lor p_{i,9,1} ) \\ &\land (p_{i,1,2} \lor p_{i,2,2} \lor p_{i,3,2} \lor p_{i,4,2} \lor p_{i,5,2} \lor p_{i,6,2} \lor p_{i,7,2} \lor p_{i,8,2} \lor p_{i,9,2} ) \\ &\land (p_{i,1,3} \lor p_{i,2,3} \lor p_{i,3,3} \lor p_{i,4,3} \lor p_{i,5,3} \lor p_{i,6,3} \lor p_{i,7,3} \lor p_{i,8,3} \lor p_{i,9,3} ) \\ &\land (p_{i,1,4} \lor p_{i,2,4} \lor p_{i,3,4} \lor p_{i,4,4} \lor p_{i,5,4} \lor p_{i,6,4} \lor p_{i,7,4} \lor p_{i,8,4} \lor p_{i,9,4} ) \\ &\land (p_{i,1,5} \lor p_{i,2,5} \lor p_{i,3,5} \lor p_{i,4,5} \lor p_{i,5,5} \lor p_{i,6,5} \lor p_{i,7,5} \lor p_{i,8,5} \lor p_{i,9,5} ) \\ &\land (p_{i,1,6} \lor p_{i,2,6} \lor p_{i,3,6} \lor p_{i,4,6} \lor p_{i,5,6} \lor p_{i,6,6} \lor p_{i,7,6} \lor p_{i,8,6} \lor p_{i,9,6} ) \\ &\land (p_{i,1,7} \lor p_{i,2,7} \lor p_{i,3,7} \lor p_{i,4,7} \lor p_{i,5,7} \lor p_{i,6,7} \lor p_{i,7,7} \lor p_{i,8,7} \lor p_{i,9,7} ) \\ &\land (p_{i,1,8} \lor p_{i,2,8} \lor p_{i,3,8} \lor p_{i,4,8} \lor p_{i,5,8} \lor p_{i,6,8} \lor p_{i,7,8} \lor p_{i,8,8} \lor p_{i,9,8} ) \\ &\land (p_{i,1,9} \lor p_{i,2,9} \lor p_{i,3,9} \lor p_{i,4,9} \lor p_{i,5,9} \lor p_{i,6,9} \lor p_{i,7,9} \lor p_{i,8,9} \lor p_{i,9,9} ) \big) \\[3pt] = \ & \big(( p_{1,1,1} \lor p_{1,2,1} \lor p_{1,3,1} \lor p_{1,4,1} \lor p_{1,5,1} \lor p_{1,6,1} \lor p_{1,7,1} \lor p_{1,8,1} \lor p_{1,9,1} ) \\ &\land (p_{1,1,2} \lor p_{1,2,2} \lor p_{1,3,2} \lor p_{1,4,2} \lor p_{1,5,2} \lor p_{1,6,2} \lor p_{1,7,2} \lor p_{1,8,2} \lor p_{1,9,2} ) \\ &\land (p_{1,1,3} \lor p_{1,2,3} \lor p_{1,3,3} \lor p_{1,4,3} \lor p_{1,5,3} \lor p_{1,6,3} \lor p_{1,7,3} \lor p_{1,8,3} \lor p_{1,9,3} ) \\ &\land (p_{1,1,4} \lor p_{1,2,4} \lor p_{1,3,4} \lor p_{1,4,4} \lor p_{1,5,4} \lor p_{1,6,4} \lor p_{1,7,4} \lor p_{1,8,4} \lor p_{1,9,4} ) \\ &\land (p_{1,1,5} \lor p_{1,2,5} \lor p_{1,3,5} \lor p_{1,4,5} \lor p_{1,5,5} \lor p_{1,6,5} \lor p_{1,7,5} \lor p_{1,8,5} \lor p_{1,9,5} ) \\ &\land (p_{1,1,6} \lor p_{1,2,6} \lor p_{1,3,6} \lor p_{1,4,6} \lor p_{1,5,6} \lor p_{1,6,6} \lor p_{1,7,6} \lor p_{1,8,6} \lor p_{1,9,6} ) \\ &\land (p_{1,1,7} \lor p_{1,2,7} \lor p_{1,3,7} \lor p_{1,4,7} \lor p_{1,5,7} \lor p_{1,6,7} \lor p_{1,7,7} \lor p_{1,8,7} \lor p_{1,9,7} ) \\ &\land (p_{1,1,8} \lor p_{1,2,8} \lor p_{1,3,8} \lor p_{1,4,8} \lor p_{1,5,8} \lor p_{1,6,8} \lor p_{1,7,8} \lor p_{1,8,8} \lor p_{1,9,8} ) \\ &\land (p_{1,1,9} \lor p_{1,2,9} \lor p_{1,3,9} \lor p_{1,4,9} \lor p_{1,5,9} \lor p_{1,6,9} \lor p_{1,7,9} \lor p_{1,8,9} \lor p_{1,9,9} ) \big) \\ &\bigwedge \big(( p_{2,1,1} \lor p_{2,2,1} \lor p_{2,3,1} \lor p_{2,4,1} \lor p_{2,5,1} \lor p_{2,6,1} \lor p_{2,7,1} \lor p_{2,8,1} \lor p_{2,9,1} ) \\ &\land (p_{2,1,2} \lor p_{2,2,2} \lor p_{2,3,2} \lor p_{2,4,2} \lor p_{2,5,2} \lor p_{2,6,2} \lor p_{2,7,2} \lor p_{2,8,2} \lor p_{2,9,2} ) \\ &\land (p_{2,1,3} \lor p_{2,2,3} \lor p_{2,3,3} \lor p_{2,4,3} \lor p_{2,5,3} \lor p_{2,6,3} \lor p_{2,7,3} \lor p_{2,8,3} \lor p_{2,9,3} ) \\ &\land (p_{2,1,4} \lor p_{2,2,4} \lor p_{2,3,4} \lor p_{2,4,4} \lor p_{2,5,4} \lor p_{2,6,4} \lor p_{2,7,4} \lor p_{2,8,4} \lor p_{2,9,4} ) \\ &\land (p_{2,1,5} \lor p_{2,2,5} \lor p_{2,3,5} \lor p_{2,4,5} \lor p_{2,5,5} \lor p_{2,6,5} \lor p_{2,7,5} \lor p_{2,8,5} \lor p_{2,9,5} ) \\ &\land (p_{2,1,6} \lor p_{2,2,6} \lor p_{2,3,6} \lor p_{2,4,6} \lor p_{2,5,6} \lor p_{2,6,6} \lor p_{2,7,6} \lor p_{2,8,6} \lor p_{2,9,6} ) \\ &\land (p_{2,1,7} \lor p_{2,2,7} \lor p_{2,3,7} \lor p_{2,4,7} \lor p_{2,5,7} \lor p_{2,6,7} \lor p_{2,7,7} \lor p_{2,8,7} \lor p_{2,9,7} ) \\ &\land (p_{2,1,8} \lor p_{2,2,8} \lor p_{2,3,8} \lor p_{2,4,8} \lor p_{2,5,8} \lor p_{2,6,8} \lor p_{2,7,8} \lor p_{2,8,8} \lor p_{2,9,8} ) \\ &\land (p_{2,1,9} \lor p_{2,2,9} \lor p_{2,3,9} \lor p_{2,4,9} \lor p_{2,5,9} \lor p_{2,6,9} \lor p_{2,7,9} \lor p_{2,8,9} \lor p_{2,9,9} ) \big) \\ &\bigwedge \dots \\ &\bigwedge \big(( p_{9,1,1} \lor p_{9,2,1} \lor p_{9,3,1} \lor p_{9,4,1} \lor p_{9,5,1} \lor p_{9,6,1} \lor p_{9,7,1} \lor p_{9,8,1} \lor p_{9,9,1} ) \\ &\land (p_{9,1,2} \lor p_{9,2,2} \lor p_{9,3,2} \lor p_{9,4,2} \lor p_{9,5,2} \lor p_{9,6,2} \lor p_{9,7,2} \lor p_{9,8,2} \lor p_{9,9,2} ) \\ &\land (p_{9,1,3} \lor p_{9,2,3} \lor p_{9,3,3} \lor p_{9,4,3} \lor p_{9,5,3} \lor p_{9,6,3} \lor p_{9,7,3} \lor p_{9,8,3} \lor p_{9,9,3} ) \\ &\land (p_{9,1,4} \lor p_{9,2,4} \lor p_{9,3,4} \lor p_{9,4,4} \lor p_{9,5,4} \lor p_{9,6,4} \lor p_{9,7,4} \lor p_{9,8,4} \lor p_{9,9,4} ) \\ &\land (p_{9,1,5} \lor p_{9,2,5} \lor p_{9,3,5} \lor p_{9,4,5} \lor p_{9,5,5} \lor p_{9,6,5} \lor p_{9,7,5} \lor p_{9,8,5} \lor p_{9,9,5} ) \\ &\land (p_{9,1,6} \lor p_{9,2,6} \lor p_{9,3,6} \lor p_{9,4,6} \lor p_{9,5,6} \lor p_{9,6,6} \lor p_{9,7,6} \lor p_{9,8,6} \lor p_{9,9,6} ) \\ &\land (p_{9,1,7} \lor p_{9,2,7} \lor p_{9,3,7} \lor p_{9,4,7} \lor p_{9,5,7} \lor p_{9,6,7} \lor p_{9,7,7} \lor p_{9,8,7} \lor p_{9,9,7} ) \\ &\land (p_{9,1,8} \lor p_{9,2,8} \lor p_{9,3,8} \lor p_{9,4,8} \lor p_{9,5,8} \lor p_{9,6,8} \lor p_{9,7,8} \lor p_{9,8,8} \lor p_{9,9,8} ) \\ &\land (p_{9,1,9} \lor p_{9,2,9} \lor p_{9,3,9} \lor p_{9,4,9} \lor p_{9,5,9} \lor p_{9,6,9} \lor p_{9,7,9} \lor p_{9,8,9} \lor p_{9,9,9} ) \big) \end{align}

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Answer to question 2:

$ \bigvee_{i = 7}^{9} p_{i} $ and $ p_{7}\vee p_{8}\vee p_{9} $ are just two different notations for the same formula (they denote the same object). So, if your question refers to notations, $\bigvee_{i = 7}^{9} p_{i}$ is the compact or implicit notation and $p_{7}\vee p_{8}\vee p_{9}$ is the expanded or explicit notation for such a formula. But if your question refers to the (same) formula denoted by $\bigvee_{i = 7}^{9} p_{i}$ and $p_{7}\vee p_{8}\vee p_{9}$, such a formula is a (disjunctive) clause, i.e. a disjunction of literals, where a literal is an atomic formula or its negation. Since it is a disjunctive clause, such a formula is both a conjunctive normal form and a disjunctive normal form.