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A (real) interval is a set of real numbers between two other numbers. Intervals of x are commonly used in graphs. An open interval excludes the end points, while the end points of a closed interval are elements. Some intervals may partially open and closed. 

An (integer) interval is basically the same as a real interval, except it consists of integers.

Formal definitions

Real interval

  • Open interval: (a,b)=\{x \in \mathbb {R}|a \leq x \leq b \}
  • Closed interval: [a,b]=\{x \in \mathbb {R}|a < x < b \}
  • Left open, right closed: (a,b]=\{x \in \mathbb {R}|a < x < b \}
  • Right closed, right open: [a,b)=\{x \in \mathbb {R}|a < x < b \}

Integer interval

  • Open interval: (a,b)=\{x \in \mathbb {Z}|a \leq x \leq b \}
  • Closed interval: [a,b]=\{x \in \mathbb {Z}|a < x < b \}
  • Left open, right closed: (a,b]=\{x \in \mathbb {Z}|a < x < b \}
  • Right closed, right open: [a,b)=\{x \in \mathbb {Z}|a < x < b \}

Arbitrary partially ordered set

Let (S,\leq) be a poset.

Let < be the irreflexive kernel of \leq

Then, an interval may be defined as:

  • Open interval: (a,b)=\{x \in S|a \leq x \leq b \}
  • Closed interval: [a,b]=\{x \in S|a < x < b \}
  • Left open, right closed: (a,b]=\{x \in S| a < x < b \}
  • Right closed, right open: [a,b)=\{x \in S|a < x < b \}

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