In my Foundations of Mathematics Textbook I encountered the following problem.
The book states that for the domain of discourse $D = \{a,b,c\}$ and binary relation defined as $E = \{(a,b),\, (a,c)\}$ the extensionality axiom is violated. What I understand this axiom to mean is that if two sets have the same elements than they are equal, so why is this true?
Edit: I believe I am confused by this notation does {a,b} mean an element obtained from a binary relation with a and b?
What extensionality actually says is that if $x$ and $y$ are such that $z\mathbin{E} x\Leftrightarrow z\mathbin{E} y$ for all $z$, then $x=y$. In your example it is true that $z\mathbin{E} b\Leftrightarrow z\mathbin{E} c$ for all $z\in D$ (since the only $z$ such that $z\mathbin{E} b$ is $z=a$ and similarly for $c$). Since $b\neq c$, this means extensionality fails.
Extensionality only says "if two sets have the same elements than they are equal" if you think of the relation $E$ as being the "element" relation $\in$ between sets. That is, if you pretended $b$ and $c$ were sets, and their elements were exactly those $z$ such that $z\mathbin{E} b$ and $z\mathbin{E} c$, then $b$ and $c$ would have exactly the same elements (namely, just $a$).