Some confusions about instrumental variable (econometrics) As the title says, I have some confusions about instrumental variable. 

According to my lecture note (which is supposed to be reliable), instrumental variable has two big properties: 


Validity: an instrumental variable and error term are not correlated and it implies that the instrumental variable does not directly lead to changes in dependent variable.
Relevance: an instrumental variable and endogenous variable are correlated. 


So here is my confusion. (let's denote z is an instrumental variable)

Endogenous variables are correlated with the error terms and z is correlated with endogenous variable. 

Doesn't this imply that z is correlated with error terms? If so, it violates the Validity (mentioned above).

In case my question isn't clear enough. Here is the case where I am stuck: $$
score_i = \beta_0 + \beta_1 course_i + u_i
$$
where $course_i$ is a binary variable of taking a preparatory course and $u_i$ is an error term.

Quite obviously, $course_i$ is correlated with $u_i$ since the parental income is one of the unobserved variables (so in $u_i$) and $course_i$ and parental income are correlated (I think this is also quite obvious). 

So far we know that parental income (I think this is an instrumental variable) is correlated with $course_i$. But since parental income was belonged to error term, it violates Validity.

Any help is welcome.
Endogenous variables are correlated with the error terms and z is
  correlated with endogenous variable.
  
  Doesn't this imply that z is correlated with error terms?


No it doesn't. For mean-centered variables for simplicity, we have for linear models,

ENDOGENEITY: $E(xu) \neq 0$  

RELEVANCE : $E(xz) \neq 0$

The above conditions do not necessarily imply non-Validity ($E(zu) \neq 0$).

A very simple example to show that we may have a valid instrument.
Assume $x, u$ are centered on their mean, $E(x) = E(u) = 0$, and that $x$ has a symmetric distribution, $E(x^3) = 0$. Assume that  $E(u \mid x) = ax$.
Then  

ENDOGENEITY

$$E(xu) = E\big[E(xu\mid x)\big] =  E\big[xE(u\mid x)\big] = aE(x^2) \neq 0.$$

Consider the candidate instrument $z = x^2 + v$, with $v$ independent from $u$, but $E(xv) \neq 0$. We have

RELEVANCE
$$E(xz) = E(x^3) + E(xv) = 0 + E(xv) \neq 0$$.

VALIDITY

$$E(zu) = E(x^2u) + E(vu) = E\big[E(x^2u\mid x)\big] + E(v)E(u)$$

$$= E\big[x^2E(u\mid x)\big] + 0 = E\big[x^2a x\big] = aE(x^3) = 0.$$