Logical Implication 4 I got stuck while reading the Discrete Mathematics book of Grimaldi. Say there are two primitives.

p: I read physics.
  q: I pass physics.

Now consider below lines.  
(p $\to$ q) $\Leftrightarrow$ ($\neg$p $\lor$ q) which can be read as "I don't read or I pass"
Now read this "I don't pass or I read". which can be written as (p $\lor$ $\neg$q). Both mean the same right? (Or only I perceived it like that?) then
(p $\to$ q) $\Leftrightarrow$ ($\neg$p $\lor$ q) $\Leftrightarrow$ (p $\lor$ $\neg$q)
But I know both are not equivalent because I have written truth table for it. But can you convince me using English language sentence (not truth tables) the above non equivalence.
 A: Your question basically is:
Why is $\neg p \lor q$ not the same as $p \lor \neg q$?
OK, take:
p: $1+1=3$
q: $1+1=2$
Then: $\neg p \lor q$ i:s '$1+1\not=3$ or $1+1=2$' .. which is true
But: $p \lor \neg q$ is: '$1+1=3$ or $1+1 \not = 2$' .. which is false
A: "I don't read or I pass".  means 1) It's possible that I don't read AND I pass and 2) It's impossible that I read AND fail. iii) Everything else is possible as either I won't read or I pass.
"I don't pass or I read"  means 1) It's POSSIBLE that I read and fail 2) It's IMPOSSIBLE that I don't read AND I pass. iii) Everything else is possible as either I won't pass or I will read.
You are thinking of the exclusive OR where Either A or B but not both.  In that case "either I don't read or I pass but not both" (So I better  read!) and "either I don't pass or I read but not both" (So I better  read!) are the same.  
But with the normal OR, where Either A or B or both can happen they arent' the same because the "both" are contradictory.
"I don't pass or I read"  (Then I better read because i might pass but if I don't I WILL fail) is not the same as "I don't read or I pass" (Then I better read because I might pass if I don't but I WILL pass if I do) are not the same.
