Consider the sets` : X =text(set of all students in your school)`, `Y =text(set of all students in your class.)`
We note that every element of `Y` is also an element of `X;` we say that `Y` is a subset of `X.` The fact that `Y` is subset of `X` is expressed in symbols as `Y sub X.` The symbol `sub` stands for `text(is a subset of)` or`text(is contained in.)`

`text(Definition :)` Set `A` is said to be a subset of a set `B` if every element of `A` is also an element of `B.` In other words, `A sub B` if whenever `a in A`, then `a in B.` It is often convenient to use the symbol `-⇒-` which means implies. Using this symbol, we can write the definition of subset as follows:
`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A sub B` if `a in A ⇒ a in B.`

We read the above statement as `-A` is a subset of `B` if `a` is an element of `A` implies that `a` is also an element of `B-.` If `A` is not a subset of `B`, we write `A ⊄ B.`
We may note that for `A` to be a subset of `B,` all that is needed is that every element of `A` is in `B.` It is possible that every element of `B` may or may not be in `A`. If it so happens that every element of `B` is also in `A`, then we shall also have `B ⊂ A.` In this case, `A` and `B` are the same sets so that we have `A ⊂ B` and `B ⊂ A ⇔ A = B,` where `-⇔-` is a symbol for two way implications, and is usually read as if and only if (briefly written as `-iff-`).

It follows from the above definition that every set `A` is a subset of itself, i.e., `A ⊂ A.` Since the empty set `phi` has no elements, we agree to say that `phi` is a subset of every set. We now consider some examples :
`(i)` The set `Q` of rational numbers is a subset of the set `R` of real numbes, and we write `Q ⊂ R.`

`(ii)` If `A` is the set of all divisors of `56` and `B` the set of all prime divisors of `56,` then `B` is a subset of `A` and we write `B ⊂ A.`

`(iii)` Let `A = {1, 3, 5}` and `B = {x : x` is an odd natural number less than `6}.` Then `A ⊂ B` and `B ⊂ A` and hence `A = B.`

`(iv)` Let `A = { a, e, i, o, u}` and `B = { a, b, c, d}.` Then `A` is not a subset of `B,` also `B` is not a subset of `A.`

`text(Properset and Superset):`
Let `A` and `B` be two sets. If `A ⊂ B` and `A ≠ B` , then `A` is called a `text(proper subset)` of `B` and `B` is called `text(superset)` of `A.`

`A = {1, 2, 3}` is a proper subset of `B = {1, 2, 3, 4}.` If a set `A` has only one element, we call it a singleton set. Thus `,{ a }` is a singleton set.


`text(Subsets of set of real numbers)`
There are many important subsets of `R`. We give below the names of some of these subsets.
The set of natural numbers `N = {1, 2, 3, 4, 5, . . .}`
The set of integers `Z = {. . ., -3, -2, -1, 0, 1, 2, 3, . . .}`
The set of rational numbers `Q = { x : x =p/q , p, q ∈ Z` and `q ≠ 0}`

`text(Intervals as subsets of R)` 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.`