I'm making a design and trying to write $P(A \text{ and } B) \leq P(A \cup B)$. I'm wondering if these are all correct:
- $P(A,B) \leq P(A \cup B)$
- $P(AB) \leq P(A \cup B)$
- $P(A \cap B) \leq P(A \cup B)$
I don't have space in the design for the third one, so I'm wondering if the first two are equivalent. I've rarely seen people writing P(AB) so I wasn't sure if it's incorrect, or more niche to e.g. computer fields evaluating boolean statements.
Thanks!
If you can somehow convey that you're using Boolean arithmetic, then you could say $P(AB) \leq P(A+B)$. It still feels weird to me but it is at least consistent (if we think of $P(A)$ as $P(A=\text{True})$). I don't like option 2 as you stated it because it seems to be mixing notations.
Option 1 also seems a bit weird to me and confused me for a bit because seeing $A,B$ made me think they were random variables. But it might be reasonable to expect people to interpret it correctly in context.