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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: $[a,b)=\{x \in S|a < x < b \}$