Why does this relation fail symmetry and transitivity properties?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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