What is this interval operation I am doing?

Question

I'm computing the interval C given the intervals A and B that makes the following true: $\forall x_c \in C$ there is a $x_a \in A$ such that $x_a + x_c \in B$.

Example: $A = [100,200]$, $B = [300,400]$, $C = [300-200,400-100] = [100,300]$.

I was wondering if this kind of calculation had a specific name (difference? distance?)

Answer 1

According to Interval arithmetic - Wikipedia:

Subtraction: $[x_1, x_2] - [y_1, y_2] = [x_1 - y_2, x_2 - y_1].$

In your notation - assuming that the condition that you state is sufficient, as well as necessary, in order for $x_c$ to belong to $C$ - we have $x_c \in C$ if and only if there exists $x_a \in A$ such that $x_a + x_c \in B.$ This condition on $x_a, x_c$ holds if and only if there exists $x_b \in B$ such that $x_a + x_c = x_b.$ That is, there exists $x_b \in B$ such that $x_c = x_b - x_a.$

That is, using Wikipedia's notation: $x_c \in C$ if and only if $x_c \in B - A$; so $C = B - A.$

As the operation is called subtraction, presumably its result can be called the difference between $A$ and $B,$ but I haven't actually got a reference for this.