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Interval functions grimoire

  1. Arithmetic operations
  2. Set operations
  3. Comparisons
  4. Scalar functions

This article is a quick reference about how different operations on intervals are defined. Unless differently stated we use the following notation

X=[x‾,x‾]Y=[y‾,y‾] X = [\underline{x}, \overline{x}]\quad Y = [\underline{y}, \overline{y}]

Arithmetic operations

OperationDefinitionX+Y[x‾+y‾,x‾+y‾]X−Y[x‾−y‾,x‾−y‾]XY[min⁡S,max⁡S] where S={x‾y‾,x‾y‾,x‾y‾,x‾y‾}1X[1x‾,1x‾] if 0∉X[−∞,∞] if x‾<0<x‾[1x‾,∞] if x‾=0≠x‾[−∞,1x‾] if x‾≠0=x‾∅ if X=[0,0]XYX⋅1Y \begin{array}{|c|c|} \hline\text{Operation}&\text{Definition}\\\hline X + Y& [\underline{x}+\underline{y}, \overline{x}+\overline{y}] \\\hline X - Y &[\underline{x}-\overline{y}, \overline{x}-\underline{y}]\\\hline XY&[\min{S}, \max{S}]\text{ where } \\&S=\{\underline{x}\underline{y}, \underline{x}\overline{y}, \overline{x}\underline{y}, \overline{x}\overline{y}\}\\\hline \frac{1}{X}&\left[\frac{1}{\overline{x}}, \frac{1}{\underline{x}}\right]\text{ if } 0 \notin X\\ & [-\infty, \infty]\text{ if }\underline{x}<0<\overline{x}\\ & \left[\frac{1}{\overline{x}}, \infty \right]\text{ if }\underline{x}=0\neq\overline{x}\\ & \left[-\infty, \frac{1}{\underline{x}}\right]\text{ if }\underline{x}\neq 0=\overline{x}\\ &\emptyset\text{ if } X = [0, 0]\\\hline \frac{X}{Y}&X\cdot\frac{1}{Y}\\\hline \end{array}

Set operations

OperationDefinitionX∩Y[max⁡(x‾,y‾),min⁡(x‾,y‾)] if max⁡(x‾,y‾)≤min⁡(x‾,y‾)∅ otherwiseX∪Y[min⁡(x‾,y‾),max⁡(x‾,y‾)] \begin{array}{|c|c|} \hline \text{Operation}&\text{Definition}\\\hline X \cap Y&[\max(\underline{x}, \underline{y}), \min(\overline{x}, \overline{y})]\text{ if }\max(\underline{x}, \underline{y})\leq \min(\overline{x}, \overline{y})\\ &\emptyset\text{ otherwise}\\\hline X \cup Y&[\min(\underline{x}, \underline{y}), \max(\overline{x}, \overline{y})]\\\hline \end{array}

Comparisons

OperationDefinitionX⋚Yx‾⋚y‾∧x‾⋚y‾X⊆Yy‾≤x‾∧x‾≤y‾ \begin{array}{|c|c|} \hline \text{Operation}&\text{Definition}\\\hline X \lesseqgtr Y&\underline{x}\lesseqgtr\underline{y}\land\overline{x}\lesseqgtr\overline{y}\\\hline X\subseteq Y& \underline{y}\leq \underline{x} \land \overline{x}\leq\overline{y}\\\hline \end{array}

Scalar functions

All the following functions return NaN if the interval is empty.

OperationDefinitioninf(X)x‾sup(X)x‾mid(X)x‾+x‾2diam(X)x‾−x‾rad(X)diam(X)2mag(X)max⁡{∣x∣:x∈X}mig(X)min⁡{∣x∣:x∈X} \begin{array}{|c|c|} \hline \text{Operation}&\text{Definition}\\\hline \text{inf}(X)&\underline{x}\\\hline \text{sup}(X)&\overline{x}\\\hline \text{mid}(X)&\frac{\underline{x}+\overline{x}}{2}\\\hline \text{diam}(X)&\overline{x}-\underline{x}\\\hline \text{rad}(X)&\frac{\text{diam}(X)}{2}\\\hline \text{mag}(X)&\max\{|x|:x\in X\}\\\hline \text{mig}(X)&\min\{|x|:x\in X\}\\\hline \end{array}
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