Why are some logical statements not immediately obvious in proofs?

[ice109]

im just starting to write proofs and it's going well but some things aren't immediately obvious to me.

for example it is not immediately obvious to me why

$$\forall_i ~ p_i \vee q_i \Leftrightarrow (\forall_i p_i ) \vee (\forall_i q_i)$$ isn't a tautology

and it wasn't immediately obvious to me why a statement like this

$$\forall_i ~ x \in A \vee B_i$$

isn't equivalent to

$$x \in A \vee \forall_i ~ x \in B_i$$

although i do understand now. can someone suggest a book or an internet resource that would help me with this? i picked up an introduction to math logic book but there's so much other stuff in there and obviously with more practice i'll get the hang of it but still some ideas on how to either get it quicker or as mentioned some resources. maybe prove a bunch of these set theorems lots of different ways.

 

[wildman]

The book "How to Prove It: A Structured Approach" by Daniel J Velleman was useful to me. The first two chapters are an easy to understand discussion of logic as it pertains to proofs.

 

[ice109]

anyone else?

 

[HallsofIvy]

Consider the statements p_i= "i is an odd number" and q_i= "i+ 1 is an odd number". Then for all i, p_i v q_i= "either i is an odd number or i+ 1 is an odd number" is true.

$\forall i p_i$, however, is the statement "for all i, i is an odd number" which is false. $\forall i q_i$ is the statement "for all i, i+ 1 is an odd number" which is also false. "false" v "false"= "false".

 

[ice109]

yea i figured that one out