Possible common misunderstanding of implication First of all I took discrete course, I know all about truth tables and stuff so I am not uninformed.I have confusion about following: 
All sources says $p\implies q$ means basically $p$ implies $q$ which is if $p$ is true then $q$ is true.It sounds ok at first but why do they assume $p\implies q$ in the first place?
$p$ can take the true value and $q$ can take the false value? Why they don't consider this and say if $p$ is true $q$ is true.
I am sure there must be logical explanation and other must have been thought it before me.
 A: You have to consider the truth of the implication as a whole. P $\implies$ Q is true, if 


*

*P is true and Q is true

*P is false (Q can be either true or false)


The only time the implication as an overall statement is false is when P is true and Q is false
See this answer:
What does "imply" mean in a statement?
A: The question is not very clear, but I'm going to make a guess at the source of your confusion and try to answer that; I apologize if my guess is wrong.  The condition "if $p$ is true then $q$ is true" is not the definition of the sequence of symbols $p\implies q$. Rather, it is the definition of under what circumstances $p\implies q$ counts as true. So the condition "if $p$ is true then $q$ is true" might well be false; it is not assumed as a premise.  It might well be the case that $p$ is true and $q$ is false, but in that situation $p\implies q$ would also be false.
A: I'm not sure this question has any informational content; you're confused about a semantic issue (I think?). The statement has to start somewhere. Why shouldn't P imply Q? Suppose $x=2$
P: x is an even number
Q: x is a multiple of 2
P implies Q because this is the way we define even numbers. I can't say the same if 
Q: x is a dog
Then, of course, P does not imply Q. There is a requirement that Q does, in fact, follow from P before you can use the statement.
A: Suppose you give a statement here in MSE "If my friend knows some Mathematics then I will join MSE". Here
$p$: My friend knows Mathematics.
$q$: I join MSE.
Then what you said is $p\rightarrow q$. Now, suppose one of the old member of MSE files a case against you and imagine that you are in the court for hearing. See the following possibilities:
$1$. Your friend knows Mathematics ($T$) and you join MSE ($T$), then you are innocent ($T$)  (as you stated nothing wrong in your statement) i.e. $T\rightarrow T\equiv T$.
$2$.Your friend knows Mathematics ($T$) and you dn't join MSE ($F$), then you are not innocent ($F$)  (as you contradicted your statement) i.e. $T\rightarrow F\equiv F$.
$3$.Your friend doesn't know Mathematics ($F$) and you join MSE ($T$), then you are innocent ($T$)  (as you never stated anything about his unawareness of Mathematics) i.e. $F\rightarrow T\equiv T$.
$4$.Your friend doesn't know Mathematics ($F$) and you do not join MSE ($F$), then you are innocent ($T$)  (as you never stated anything about his unawareness of Mathematics) i.e. $F\rightarrow F\equiv T$.
Note: You cannot be punished for what you never claimed or stated. Consider another innocent statement, "If I am the creator of this world then elephants fly in the sky".
