In classical propositional logic, or (aka. disjunction) is a Boolean function $\vee:\{0,1\}^2\longrightarrow\{0,1\}$ defined as $\vee:=\{ (00,0), (01,1), (10,1),(11,1) \}$, and and (aka. conjunction) is a Boolean function $\wedge: \{0,1\}^2\longrightarrow\{0,1\}$ defined as $\{(00,0),(01,0),(10,0),(11,1)\}$.
There's also rules of inference for these functions, such as $\vee$-introduction, $\wedge$-introduction, $\vee$-elimination, and $\wedge$-elimination, as well as other machinery from logic.
So, or and and are not the same object. But when using them in math reasoning, sometimes the 2 can be used interchangeably.
Example. Let $X$ be a topological space, let $A\subseteq X$, let $\text{Lim }A$ be the set of limit points of $A$. The closure of $A$ is defined as $A\cup\text{Lim }A$. But the set $A\cup \text{Lim }A$ is defined as the set $\{x\in X\ |\ x\in A\ \vee\ x\in \text{Lim }A\}$, namely, the set of all $x\in X$ where $x\in A$ or $x\in \text{Lim }A$. Equivalently, th closure of $A$ is the set $A$ and its limit points.
(This is the first example that came to mind, but hopefully you get the idea.)
I know I'm mixing formal logic with natural (human!) language to the point of frivolity, but I'm wondering if there's a non-mundane reason for this equivalence. Perhaps or and and are "dual", in some sense? Or perhaps the equivalence arises just because natural language is really weird?
(One could disregard this question by saying "the meaning of or in logic is not the same as in natural language" (usually natural language uses exclusive-or), just like the meaning of if in logic is not the same as in natural language (usually natural language uses if-and-only-if), but I get the feeling something else is going on.)
Best Answer
Consider: "All fruits and vegetables are nutritious"
Rather than:
$$\forall x ((F(x) \land V(x)) \rightarrow N(x)) \text{ Wrong!}$$
it translates as:
$$\forall x ((F(x) \lor V(x)) \rightarrow N(x))$$
But it also translates as (and is indeed equivalent to):
$$\forall x (F(x)\rightarrow N(x))\land \forall x (V(x) \rightarrow N(x))$$
So now we see that "Fruits and vegetables are nutritious" is really just shorthand for "fruits are nutritious and vegetables are nutritious"
Applied to your case:
"the closure of A is the set A and its limit points."
can be translated as:
$$\forall x ((x \in A \lor x \in \text{Lim} A) \rightarrow x \in Closure(A))$$
or as:
$$\forall x (x \in A \rightarrow x \in Closure(A)) \land \forall x (x \in \text{Lim} A) \rightarrow x \in Closure(A))$$
In other words, the confusion is because of the difference between the disjunction of conditions and conjunction of conditionals.