Prove that for any directed graph G = (V, E), the following inequality holds: d(A) + d(B) ≥ d(A ∩ B) + d(A ∪ B)

Consider the disjoint subsets $A\cap B, A-B, B-A, S-A - B$.
Consider an edge that goes from one subset to another distinct subset.
How many times is this edge counted on the LHS?
How many times is this edge counted on the RHS?

Is the first number always larger than or equal to the second number?

IE An edge from $A-B$ to $B-A$ is counted in $d(A)$ but not in $ d(B)$, $d( A \cap B)$, or $d(A \cup B)$, so the edge is counted at least as many times on the LHS as it is on the RHS.
If we can show this for each of these types of edges, then the inequality holds.

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Why an initial ring is a domain?

Calculate the area of an triangle that is inscribed into an ellipse such that the elliptical sectors are of equal size.

Equality of left and right inverses in a finite domain

Show that for all $X \in \mathscr{M}_n(\mathbb{C})$ we have that: $\lim\limits_{n \rightarrow \infty}{\left(I + \frac{1}{m}X\right)}^{m} = \exp{X}$

Suppose the equation of motion of a material point is given by the equation $s(t)=e^t$. Does it follow that $s(t)=v(t)=a(t)=e^t$?

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{f(x) belongs to R[x] | f(n) belongs to Z for all n belongs to Z}uncountable or countable ? (TIFR GS 2022) [closed]

Unnecessary assumption in Roman's Advanced Linear Algebra, exercise 3.10?