use set builder notation to describe the set of:
a.) all odd numbers between 100 and 200
b.) all points on the graph of the function y=x^2

a) { x : x = 2n + 1 ∀ n ∈ N where 50 ≤ n ≤ 99 }

b) {(x,y) : ∀ x ∈ R, y∈ R}

Step-by-step explanation:

Set builder form of a set is representation of the set in symbol and word inside {}.

a) All odd numbers between 100 and 200,

∵ every odd number is in the form of 2n + 1 where n ∈ N,

Also, the odd numbers between 100 and 200 are,

101, ....199

If 2n + 1 = 101

⇒ 2n = 100 ⇒ n = 50

If 2n + 1 = 199

⇒ 2n = 198 ⇒ n = 99

Then the set would be,

{ x : x = 2n + 1 ∀ n ∈ N where 50 ≤ n ≤ 99 }

b) All points on the graph of the function ,

Since, both range and domain of the function are 'Set of all real numbers'

⇒ x ∈ R, y ∈ R,

Also, (x,y) that shows the relation will belong to the function,

Hence, the required set builder form would be,

{(x,y) : ∀ x ∈ R, y∈ R}

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation: