# Formalizing Venn diagram probability calculations

I've just started learning about using sets with probability - in the exercises I was given I can 'intuitively' use Venn diagrams to calculate the answers but I am struggling to 'formalize' my solution with the rules we've learned.

For example, if I have three intersecting events A, B & C, I can see that:

$$P(A∩B^c∩C^c) =P(A)−P(A∩B)−P(A∩C)+P(A∩B∩C)$$

but I have no idea how to formally show this. How can I prove this, and how can I move from using Venn diagrams to the formal rules?

A tedious but systematic way forward would be to use the rule $P(X)=P(X\cap Y)+P(X\cap Y^c)$ repeatedly until every term on both sides of the equation is the probability of an intersection of either $X$ or $X^c$ for each free set $X$ in the identity.
In your example, $P(A)$ becomes $$P(A\cap B\cap C)+P(A\cap B^c\cap C)+P(A\cap B\cap C^c)+P(A\cap B^c\cap C^c)$$ and so forth. Then collecting like terms and simplifying should allow everything to cancel out.