`\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].`

`"Definition :"` The collection of all subsets of a set `A` is called the power set of `A.`
• It is denoted by `P(A).` In `P(A)`, every element is a set.

E.g. If `A = { 1, 2 }`, then `P( A ) = { Phi ,{ 1 }, { 2 }, { 1,2 }}` Also, note that `n [ P (A) ] = 4 = 2^2`

In general, if `A` is a set with `n(A) = m`, then it can be shown that `n [ P(A)] = 2^m`

• Universal Set is the set defines as the set containing all objects or elements and of which all other sets are subsets.

or

• A universal set is the collection of all objects in a particular context or theory. All other sets in that framework constitute subsets of the universal set.

• Universal Sets are usually named with a capital letter. Therefore, the universal set is usually named with the capital letter `U.`

For example, R can be set of real numbers.