This is a simple logic problem, I don't know why so many people struggle with it.
Atheism literally means "without theism". An atheist is someone who does not accept, as true, claims that assert the existence of gods.
Considering the claims regarding the existence of a god, there are two possible claims:
- 1. God exists
- 2. God does not exist
For either claim, there are two positions one can take with regard to belief:
- 1. belief or acceptance of the claim
- 2. disbelief or rejection of the claim
For claim number 1 (God exists), the theist's position is one of belief, while the atheist's position is one of disbelief. For claim number 2 (God does not exist), the theist's position is one of disbelief, while atheists can hold either position.