Question

Describe the following sets in set-builder form :

(i) A = {1,2,3,4,5,6}

(ii) B = {1, $\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{4},......$}

(iii) C = {0,3,6,9,12,.....}

(iv) D = {10,11,12,13,14,15}

(v) E = {0} (vi) {1,4,9,16,...., 100}

(vii) {2,4,6,8, ....}

(viii) {5, 25, 125, 625}

Answer 1

(i) In set Builder form, a set is described by some characterizing property p(x) of its elements x.

In this case a set can be described as {x : p(x) hold} or

{x | P(x) holds} which is read as 'the set of all x such that P(x) holds'.

The symbols ':' or 'I' is read as 'such that'.

So, the above set A in Set-Builder form may be written as

$A=XϵN:x<7$

i.e. A is the set of natural numbers x such that x is less than 7.

Or

A ={ $XϵN|\left(\frac{N}{2}\underset{–}{<}x\underset{–}{<}6\right)1\underset{–}{<}x\underset{–}{<}6$ }

i.e. A is the set of natural numbers x such that x is greater than or equal 1 and less than or equal to 6.

(ii) In Set Builder form, a set is described by some characterizing property P (x) of its elements x.

In this case a set can be described as {x : P (x) hold } or

{x | P (x) holds} which is read as 'the set of all x such that P (x) holds'.

The symbols ' : ' or ' I ' is read as 'such that'.

B= {$x:x=\frac{1}{n},nϵN$ }

i.e. B is the set of all those x such that $x=\frac{1}{n}$, where $nϵN$

(iii) In set Builder form, a set is described by some characterizing property P (x) of its element x.

In this case a set can be described as {x : P (x) hold } or

{x | P(x) holds} which is read as 'the set of all x such that P(x) holds'.

The symbols ':' or 'I' is read as 'such that'.

C={ $x:x=3k,kϵZ^{+}$ , the set of positive integers}

i.e. C is the set of multiplies of 3 including 0

(iv) In set Builder form, a set is described by some characterizing property P (x) of its element x.

In this case a set can be described as {x : P (x) hold } or

{x | P (x) holds} which is read as 'the set of all x such that P (x) holds '.

The symbols ':' or 'I' is read as 'such that'.

$D=\{xϵN:9<x<16\}$

i.e. D is the set of natural numbers which are more than 9 but less than 16.

(v) In Set Builder form, a set is described by some characterizing property P (x) of its element x.

In this case a set can be described as {x : P (x) hold} or

{x | P (x) holds} which is read as 'the set of all x such that P (x) holds'.

The symbols ':' , or 'I' is read as 'such that'.

$E=\{xϵZ:-1<x<1\}$

Or

E ={$xϵZ:x=0$ }

(vi) In set Builder form, a set described by some characterizing property P (x) of its element x.

In this case a set can be described as {x : P (x) hold} or

{x | P (x) holds} which is read as 'the set of all x such that P (x) holds'.

The symbols ':' or 'I' is read as 'such that'.

As $1^2=12^2=43^2=9::{10}^2=100$

$\therefore$ The above set may be described as

{$x^2:xϵN1\underset{–}{<}x\underset{–}{<}10$}

(vii) In set Builder form, a set is described by some characterizing property P (x) of its elements x.

In this case a set can be described as {x : P (x) hold } or

{x | P (x) holds} which is read as 'the set of all x such that P (x) holds'.

The symbols ':' or 'I' is read as 'such that'.

The given set can be described as

{ $x:x=2n,nϵN$} {$\therefore$ 2,4,6, .... are multiples of 2}

(viii) In set Builder form, a set is described by some characterizing property P (x) of its element x.

In this case a set can be described as {x : P (x) hold} or

{x | P (x) holds} which is read as 'the set of all x such that P (x) holds'.

The symbols ':' or 'I' is read as 'such that'.

$\therefore$

$5^1=55^2=255^3=1255^4=625$

$\therefore$ The above set can be described as

{ $x:x=5^n,1\underset{–}{<}n\underset{–}{<}4$}