Intervals

Summary

• In this section, \(a<b\)
• \(\left( a,b \right)\) is called the open interval from \(a\) to \(b\) . \(\left( a,b \right)\) is defined as the subset of \(R\) consisting of all real numbers greater than \(a\) but less than \(b\) . Formally,

\[\left( a,b \right)=\{ x∈R \right| a<x<b}\]

• \(\left[ a,b \right]\) is called the closed interval from \(a\) to \(b\) ,

\[\left[ a,b \right] =\{ x∈R \right| a≤x≤b}\]

• \(\left( a,b \right]\) is called the interval half open form the left from \(a\) to \(b\) ,

\[\left( a,b \right] =\{ x∈R \right| a<x≤b}\]

• \(\left[ a,b \right)\) is called the interval half open form the right from \(a\) to \(b\) ,

\[\left[ a,b \right) =\{ x∈R \right| a≤x<b}\]

• \(\left( -∞,b \right),\left( -∞,b \right],\left( a,∞ \right), \left[ a,∞ \right)\) are examples of unbounded intervals (the previous four are bounded intervals). For example,

\[\left( a,∞ \right)=\{ x∈R \right| x>a}\]

• \(\left( -∞,∞ \right)\) is the same set as \(R\)
• Boundary points :
• \(a,b\) are boundary points of \(\left[ a,b \right]\)
• \(b\) is a boundary point of \(\left( a,b \right]\)
• \(a\) is a boundary point of \(\left[ a,b \right)\)
• \(\left( a,b \right)\) has no boundary points
• The interior of the intervals \(\left( a,b \right), \left[ a,b \right], \left( a,b \right], \left[ a,b \right)\) is the open interval \(\left( a,b \right)\)
• The closure of the intervals \(\left( a,b \right), \left[ a,b \right], \left( a,b \right], \left[ a,b \right)\) is the closed interval \(\left[ a,b \right]\)