Why does this relation fail symmetry and transitivity properties? The question states, let $S$ be the set of all humans.
Define $a ∼ b$ iff $a$ is a full-brother
of $b$. Symmetry: Since $a$ shares both parents with $b$, then $b$ shares both parents with $a$. Would this be false because $b$ is not defined as a male, so $b$ is instead the full sister of $a$? Transitivity: Since $a$ shares both parents with $b$, and $b$ shares both parents with $c$, then a shares both parents with $c$. What does the $c$ mean in this context? Is it simply another person introduced?
 A: I am a full brother of my sister, but my sister is not a full brother of me.  So this relation is not symmetric.
EDIT: Transitivity fails: see fleablood's comment.
A: Your title is inaccurate.  An equivalence relationship can't fail symmetric and  transitive properties, by definition, and this is not an equivalence relation because it does fail.
It fails reflexive because $a $~$a $ never happens.  No-one is their own brother.
It fails symmetry for exactly the reason you state. If Allen, a boy, and Betty, girl, have the same parents than Allen is a full brother to Betty, but Betty is not a full brother to Allen.
Update!
Transitivity fails.  If Allen is a full brother to Bob.  And Bob is a full brother to Allen then Allen is not a full brother to Allen.  
Transitivity fails.  
A: For a relation to be equvalence relation you also need reflexivity that is 
$$
a\sim a, \qquad \forall a \in S.
$$ 
which would mean that $a$ is a full brother of himself which is absurd.
Reflecting on your other questions if you define $\sim$ to be brothership then you definitely run into trouble with different sexes. So in case of $a$ and $b$ has different sex it is not holding up. 
For the question about transitivity you would read, "a is a full brother of b" and "b is a full brother of c". I would say that this holds up.
I hope I could help
EDIT:
According to the comments below not even transitivity is fullfilled
