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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?

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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.

This corresponds directly to keeping track of how many times each primitive area in the Venn diagram is added and subtracted on each side of the equals sign.

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