`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)`
Let `a, b ∈ R` and `a < b.`
• Then the set of real numbers `{ y : a < y < b}` is called an `"open interval"` and is denoted by `(a, b).`
• All the points between `a` and `b` belong to the open interval `(a, b)` but `a, b` themselves do not belong to this interval.
• The interval which contains the end points also is called `"closed interval"` and is denoted by `[ a, b ].` Thus
` \ \ \ \ \ \ \ \ \ [ a, b ] = {x : a ≤ x ≤ b}`
• We can also have intervals closed at one end and open at the other, i.e.,
`[ a, b ) = {x : a ≤ x < b}` is an open interval from `a` to `b`, including `a` but excluding `b.`
`( a, b ] = { x : a < x ≤ b }` is an open interval from `a` to `b` including `b` but excluding `a.`
These intervals are also known as semi open or semi closed or half open or half closed.
`[a,b)` is also written as ` [a,b[`
`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)`
`(a,b]` is also written as ` ]a,b]`
• These notations provide an alternative way of designating the subsets of set of real numbers.
For example , if `A = (–3, 5)` and `B = [–7, 9],` then `A ⊂ B.`
On real number line, various types of intervals described above as subsets of `R,` are shown in the Fig.
• Here, we note that an interval contains infinitely many points.
The number `(b – a) ` is called the length of any of the intervals `(a, b), [a, b], [a, b)` or `(a, b].`
`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)`
Let `a, b ∈ R` and `a < b.`
• Then the set of real numbers `{ y : a < y < b}` is called an `"open interval"` and is denoted by `(a, b).`
• All the points between `a` and `b` belong to the open interval `(a, b)` but `a, b` themselves do not belong to this interval.
• The interval which contains the end points also is called `"closed interval"` and is denoted by `[ a, b ].` Thus
` \ \ \ \ \ \ \ \ \ [ a, b ] = {x : a ≤ x ≤ b}`
• We can also have intervals closed at one end and open at the other, i.e.,
`[ a, b ) = {x : a ≤ x < b}` is an open interval from `a` to `b`, including `a` but excluding `b.`
`( a, b ] = { x : a < x ≤ b }` is an open interval from `a` to `b` including `b` but excluding `a.`
These intervals are also known as semi open or semi closed or half open or half closed.
`[a,b)` is also written as ` [a,b[`
`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)`
`(a,b]` is also written as ` ]a,b]`
• These notations provide an alternative way of designating the subsets of set of real numbers.
For example , if `A = (–3, 5)` and `B = [–7, 9],` then `A ⊂ B.`
On real number line, various types of intervals described above as subsets of `R,` are shown in the Fig.
• Here, we note that an interval contains infinitely many points.
The number `(b – a) ` is called the length of any of the intervals `(a, b), [a, b], [a, b)` or `(a, b].`