`star` Subsets , Proper subset and Superset

`star` Intervals as subsets of R

`star` Intervals as subsets of R

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.

• 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.

Q 3087367287

Consider the sets

`phi, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.`

Insert the symbol ⊂ or ⊄ between each of the following pair of sets:

(i) `phi . . . B` (ii) `A . . . B` (iii) `A . . . C` (iv) `B . . . C`

`phi, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.`

Insert the symbol ⊂ or ⊄ between each of the following pair of sets:

(i) `phi . . . B` (ii) `A . . . B` (iii) `A . . . C` (iv) `B . . . C`

(i) `phi ⊂ B` as `phi` is a subset of every set.

(ii) `A ⊄ B` as `3 ∈ A` and `3 ∉ B`

(iii) `A ⊂ C` as `1, 3 ∈ A` also belongs to C

(iv) `B ⊂ C` as each element of B is also an element of C.

Q 3087467387

`A = { a, e, i, o, u}` and `B = { a, b, c, d}`.

(1) . Is A a subset of B ?

(2) . Is B a subset of A?

(1) . Is A a subset of B ?

(2) . Is B a subset of A?

(1) . No

(2) . No

Q 3017567480

Let A, B and C be three sets. If `A ∈ B` and `B ⊂ C,` is it true that `A ⊂ C?`. If not, give an example.

No. Let `A = {1}, B = {{1}, 2}` and `C = {{1}, 2, 3}.` Here `A ∈ B` as `A = {1}` and `B ⊂ C`. But `A ⊄ C` as `1 ∈ A` and `1 ∉ C`.

Note that an element of a set can never be a subset of itself.

`\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].`

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].`

Q 1904145958

Write the following as intervals :

(i) `{x : x in R , - 4 < x le 6}`

(ii) `{x : x in R, - 12 < x < -10}`

(iii) `{x : x in R, 0 le x < 7}`

(iv) `{x : x in R, 3 le x le 4}`

Class 11 Exercise 1.3 Q.No. 6

(i) `{x : x in R , - 4 < x le 6}`

(ii) `{x : x in R, - 12 < x < -10}`

(iii) `{x : x in R, 0 le x < 7}`

(iv) `{x : x in R, 3 le x le 4}`

Class 11 Exercise 1.3 Q.No. 6

(i) `(-4,6]`

(ii) `(-12,-10 )`

(iii) `[0 , 7)`

(iv) `[3,4]`