Intervals

Summary

  • In this section, a<ba<b
  • (a,b)(a,b) is called the open interval from aa to bb . (a,b)(a,b) is defined as the subset of RR consisting of all real numbers greater than aa but less than bb . Formally,

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

  • [a,b][a,b] is called the closed interval from aa to bb ,

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

  • (a,b](a,b] is called the interval half open form the left from aa to bb ,

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

  • [a,b)[a,b) is called the interval half open form the right from aa to bb ,

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

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

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

  • (−∞,∞)(−∞,∞) is the same set as RR
  • Boundary points :
    • a,ba,b are boundary points of [a,b][a,b]
    • bb is a boundary point of (a,b](a,b]
    • aa is a boundary point of [a,b)[a,b)
    • (a,b)(a,b) has no boundary points
  • The interior of the intervals (a,b),[a,b],(a,b],[a,b)(a,b),[a,b],(a,b],[a,b) is the open interval (a,b)(a,b)
  • The closure of the intervals (a,b),[a,b],(a,b],[a,b)(a,b),[a,b],(a,b],[a,b) is the closed interval [a,b][a,b]