Consider the sets : `X =` set of all students in your school, `Y =` 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 ⊂ X.` The symbol `⊂` stands for `"‘is a subset of’"` or `"‘is contained in’."`

`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT" `
A 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 ⊂ B` if whenever `a ∈ A,` then `a ∈ 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 subset B` if `a ∈ A ⇒ a ∈ 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`”.

`\color{blue} (✍️ "If A is not a subset of B, we write A ⊄ B")`.

`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT) `
• It follows from the above definition that every set A is a subset of itself, i.e., `A subset A`.
• Since the empty set `φ` has no elements, we agree to say that `φ` 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.`

`\color{green} (✍️ "Proper subset and Superset")`
Let `A` and `B` be two sets. If `A ⊂ B` and `A ≠ B` , then `A` is called a proper subset of `B` and `B` is called superset of `A.`

For example, `A = {1, 2, 3}` is a proper subset of `B = {1, 2, 3, 4}.`

`\color{green}( ✍️ "Singleton set: ")`
If a set `A` has only one element, we call it a singleton set. Thus ,`{ a }` is a singleton set.

`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)`
Let `a, b ∈ R`

• `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.

`\color{blue} (✍️ "The number (b – a) is called the length of any of the intervals")` `(a, b), [a, b], [a, b)` or `(a, b].`