It comes from boolean logic, not from set theory (which is what it’s used for here).
A implies B means that “If A is true, B is also true”, so there’s the implication. “If the thing we are talking about is a desk, it is also a piece of furniture”.
The implication has no effect if A is not true: “If the thing we are talking about is not a desk, that doesn’t say anything about whether it’s furniture or not”.
So the implication is only false if A is true and B is false. In any other case the implication is satisfied and the result is true.
Set theory just took over all of the boolean logic functions, since they work exactly the same in set theory, so they also took over the naming even if the name “implies” doesn’t really make much sense in set theory.
It comes from boolean logic, not from set theory (which is what it’s used for here).
A implies B means that “If A is true, B is also true”, so there’s the implication. “If the thing we are talking about is a desk, it is also a piece of furniture”.
The implication has no effect if A is not true: “If the thing we are talking about is not a desk, that doesn’t say anything about whether it’s furniture or not”.
So the implication is only false if A is true and B is false. In any other case the implication is satisfied and the result is true.
Set theory just took over all of the boolean logic functions, since they work exactly the same in set theory, so they also took over the naming even if the name “implies” doesn’t really make much sense in set theory.