Axler's exercise 2.17 states the following.
You might guess, by analagy with the formula for the number of elements in the union of three subsets of a finite set, that if $U_1$, $U_2$, $U_3$ are subspaces of a finite dimensional vector space, then \begin{align*} \dim(U_1 + U_2 + U_3) & = \dim U_1 + \dim U_2 + \dim U_3 \\ & - \dim(U_1 \cap U_2) - \dim(U_1 \cap U_3) - \dim(U_2 \cap U_3) + \dim(U_1 \cap U_2 \cap U_3). \end{align*} Prove or give a counterexample.
I know for a fact that this statement is false and I don't have much trouble proving a counterexample. But I always have difficulty reasoning a-priori as to why a statement like this is true or false, which requires some level of intuition. When I look at this statement, I see the principle of inclusion-exclusion, which would lead me to the conclusion that it's true. I'm having difficulty understanding the intuition for why it's false.
Could anyone lead me in the right direction?