I keep seeing written in texts the phrase 'identify sets', for example:
Identify $A$ as a subset of $F(A)$ by $a\mapsto f_a$, where $f_a$ is the function which is 1 at a and 0 elsewhere.
This is in the construction of a free module on the set A, but I don't think that's important - what is important is that A is not a subset of F(A) in the sense I'm used to.
Another example is in the construction of the tensor algebra $T(V)=\bigoplus _{n\in \textbf{N}_0}T^nV$, where we 'identify' the k-fold tensor product $T^kV$ as a subset of $T(V)$ through an inclusion map.
Please can someone explain to me what this term 'identify' means in the context of sets?
Thanks
These texts are announcing that they'll be conflating $x$ and $i(x)$, where $i\colon X\to Y$ is an injective function, and asking you to go along with it. You're right that it's $i(X)$ which is the subset of $Y$, not $X$, but sometimes there's not much harm in pretending they're the same object, and doing so might reduce clutter in your notation.
(Another typical example is the identification of $\mathbb R^n\oplus\mathbb R^m$ with $\mathbb R^{n+m}$. Technically they're different — the elements of $\mathbb R^{n+m}$ are $(n+m)$-tuples, but the elements of $\mathbb R^n\oplus\mathbb R^m$ are pairs whose first elements are $n$-tuples and whose second elements are $m$-tuples — but there's an obvious isomorphism and, often, giving that isomorphism a name and being careful to use it explicitly whenever appropriate is a lot of care for not much benefit.)