Answer
1)
The truth value for $\vee_{i=1}^{100}\left(p_{i} \wedge p_{i+1}\right)$ is false.
The reason is: since there is one even integer and one odd integer among $i$ and $i+1$, the truth value of $p_{i} \wedge p_{i+1}$ is always false, and so this is an
or-statement of all false statements, which is false.
2)
The truth value for $\bigwedge_{i=1}^{100}\left(p_{i} \vee p_{i+1}\right)$ is true.
The reason is : since there is one even integer and one odd integer among $i$ and $i+1$, the truth value of $p_{i} \vee p_{i+1}$ is always true, and so this is an
and-statement of all the true statements, which is true.
Work Step by Step
1)
The truth value for $\vee_{i=1}^{100}\left(p_{i} \wedge p_{i+1}\right)$ is false.
The reason is: since there is one even integer and one odd integer among $i$ and $i+1$, the truth value of $p_{i} \wedge p_{i+1}$ is always false, and so this is an
or-statement of all false statements, which is false.
2)
The truth value for $\bigwedge_{i=1}^{100}\left(p_{i} \vee p_{i+1}\right)$ is true.
The reason is : since there is one even integer and one odd integer among $i$ and $i+1$, the truth value of $p_{i} \vee p_{i+1}$ is always true, and so this is an
and-statement of all the true statements, which is true
