By means of an example, show that $P(A) + P(B) = 1$ does not mean that $B$ is the complement of $A$ I'm in grade 10, and I've just started to learn about complementary events. I am rather perplexed with this question. Isn't this question kinda contradictory, since $P(A) + P(A') = 1$?
This is what I got to:
$P(A) + P(B) = 1$
$P(A) + P(A') = 1$
How could it be proven that $B$ isn't the complement of $A$?
Help would be greatly appreciated.

Let $S$ be any infinite set and let $e \in S$. Now let $A$ and $B$ be a partition of $S \setminus \{e\}$ with both $A$ and $B$ infinite. Let $P(C)$ be the probability of drawing randomly an element of $S$ that is in the set $C$.
Since $S = A \cup B \cup \{e\}$ we have:
$$
P(A) + P(B) + P(\{e\}) = 1
$$
but also
$$
P(\{e\}) = 0
$$
since it is finite, thus:
$$
P(A) + P(B) = 1
$$
For example $S = \mathbb{N}$, $e=\{0\}$, $A=\{x \mid x \in \mathbb{N}_0 \wedge x \text{ is even}\}$ and $B=\{x \mid x \in \mathbb{N}_0 \wedge x \text{ is odd}\}$.
