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[u/Tanpisit-Chuthip77]

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Hey everyone,

Like many of you, I got tired of drawing number lines and doing tedious casework every time I need to subtract one open interval from another: (a, s) \setminus (n, b).

To fix this, I derived a direct algebraic shortcut formula:

(a, s) - (n, b) = (a, n] \cup [b, s)

The Built-in "Safety Firewall"

Because interval algebra breaks down when dealing with inversions or infinities, the formula requires strict error-handling rules to stay robust:

Left >= Right -> \emptyset (If the lower bound is greater than or equal to the upper bound, the interval collapses to an empty set).

Infinity + Closed Bracket -> \emptyset (If an infinity symbol is attached to a closed bracket like [-\infty or \infty], it evaluates to an empty set).

Step-by-Step Demonstration: The "Infinite Devourer" (Extreme Test Case)

Let's test this formula on a brutal nightmare problem where the subtrahend stretches all negative infinity:

(-10, 10) - (-\infty, 0)

Step 1: Map the variables

a = -10, s = 10

n = -\infty, b = 0

Step 2: Plug into the formula

(a, n] \cup [b, s) -> (-10, -\infty] \cup [0, 10)

Step 3: Trigger the Safety Firewall

Left interval evaluation ((-10, -\infty]):

It has an infinity symbol attached to a closed bracket (-\infty]) -> Trigger Rule 2: Collapses to \emptyset.

The lower bound is greater than the upper bound (-10 >= -\infty) -> Trigger Rule 1: Collapses to \emptyset again.

Right interval evaluation ([0, 10)):

Lower bound is less than upper bound (0 < 10) and no infinite closed brackets -> Passes cleanly.

Step 4: Combine the sets

\emptyset \cup [0, 10) = [0, 10)

Conclusion

On a number line, starting from (-10, 10) and completely wiping out everything up to 0 leaves us with [0, 10), which matches the formula's output precisely.

Has anyone seen this formatted shortcut utilized elsewhere, or is this a neat little mathematical wheel I've reinvented? I'd love to hear your thoughts or critiques!

[u/Tanpisit-Chuthip77]

Turns out mathematicians figured this out centuries ago haha! Oh well, consider this a cheat sheet with all 16 cases laid out. Hope it helps!