I don't understand how implication works in logic The statement $(p \wedge q ) \Rightarrow (p \Rightarrow q)$ is a tautology, i.e., always true.
Now, imagine I am writing a proof and I want to show that $p \Rightarrow q$. I show that statement $p$ is always true, then I show that statement $q$ is always true, then I can end my proof and claim $p \Rightarrow q.$
This doesn't make sense to me. If I prove that $\sqrt2$ is irrational and then I prove that $7$ is odd, I don't understand how "$\sqrt2$ is irrational" $\Rightarrow$ "$7$ is odd".
Can someone help this disconnect I am having? Perhaps, you can provide information about what these logical statements actually mean and how they are used.
 A: Here is an example, outside of mathematics, that I think shows how material implication is accepted and understood in ordinary human discourse. There was a court case, in which the wife of a serial killer contested claims, made in the satirical magazine Private Eye, that she had tried to profit on her husband's crimes. When, she won the case, Ian Hislop, the editor of the the magazine stated that "if that's justice, I am a banana". He was quite right and she ended up the poorer for it.
A: Note that logical implication (aka ‘material conditional’) does not translate directly into our intuitive understanding of implication. We often think of implication as synonymous with causality, whereas the material conditional exists irrespective of causality. Mathematics isn’t a science, and its truths are almost always absolute (relative to the chosen axioms), which makes mathematical truth rather than causation its object of study. You can try and shift your thinking in this direction. Try and think of statements like $a\implies b$ as “if $a$ is true, then $b$ also happens to be true” rather than “$a$ causes $b$”.
Note that the main instrument of mathematics is proof, and as used in proofs, the material implication’s purpose is not to formalise some causal connection between mathematical facts, but rather to verify that one of the statements is indeed a fact. Implications of the sort you mention, where the consequent is tautologically true, are of little interest to mathematicians, even though they are correct owing to the truth-functional definition of the material conditional.
To help you come to terms with this definition, remember that the only situation where $a\implies b$ fails to be true is when $a$ is true and $b$ is false. When doing proofs and saying that $b$ follows from $a$, what we really mean is that whenever $a$ is true, $b$ cannot possibly be false (hence is also true). The material conditional guarantees that. And it is that guarantee that is its sole purpose, and what makes making logical deductions in proofs at all possible.
A: The meaning of the conditional $\implies$ has a different meaning than in ordinary language.
$\sqrt{2}$ is irrational $\implies$ 7 is odd
Is a true statement precisely for the reasons you claimed. In ordinary language we use "if-then" or $P \implies Q$ statements almost exclusively in situations where there is some set of "distribution" over $P$; that is, $P$ can be true or false, and then we ask whether or not in every situation where $P$ is true does it also follow that $Q$ is true. There is also often a notion of "causality."
Again; crucially, in normal language when we say $P \implies Q$ there must be at least two possible scenarios: one in which $P$ could be true and one in which $P$ could be false. Furthermore, things get a little murkier here, but $P$ being true must be the "reason" that $Q$ is true.
The fact that this is the way the statement gets used in ordinary language is why the mathematical statement feels odd. Modal logic can more closely mirror the "natural language" usage. https://en.wikipedia.org/wiki/Modal_logic
A: This is one of the biggest issues with the conditional connective in propositional logic. This problem arises from the fact  that $p\rightarrow q\equiv \neg p\vee q$. All of the logical connectives in the propositional calculus produce a sentence whose truth value is only determined by the truth values of the propositions it connects. In general when we think or say $p$ implies $q$ we are also implies some form of causation which is not captured by the logical connective $\rightarrow$. This means that $\sqrt(2)\notin \mathbb{Q}\rightarrow 7\text{ is odd}$. is a true sentence even though there seems to be no real connection between the two things being true.
A: 
I show that statement $p$ is always true, then I show that statement
$q$ is always true, then I can end my proof and claim $p \Rightarrow q.$ This doesn't make sense to me.

Using a simple form of natural deduction, we have:
(Screen shot from my proof checker)

