Question

Roster method and Set builder notation

1. Use the roster method to list the members of these sets.

A = { y | y is the square of an integer and y < 100}

B = {x ∈ ℚ | x = a / b, a = 3, b ∈ ℤ⁺ , x ≥ 2 }

C = { y | y is the remainder of x/5, x ∈ ℤ⁺ } another definition: C = {y | y = x%5, x∈ ℤ⁺ }

Answer 1

A = { y | y is the square of an integer and y < 100}

Let consider y starts from 1 and y<100

so set contains 1,4,9,16,25,36,49,64,81

Hence in roster form A={1,4,9,16,25,36,49,64,81}

B = {x ∈ ℚ | x = a / b, a = 3, b ∈ ℤ⁺ , x ≥ 2 }

z+ means positive integers and given x>=2, so x=a/b (>=2)

if b=1 , x=3/1 (>=2) ---->taken

if b=2 , x=3/2 (<2) -----> not taken

if b=3 , x=3/3 (<2) ------> not taken

if b=4, x=3/4 (<2) --------> not taken

if b=5, x=3/5(<2)-------> not taken

...

so only possible value of b is 1 , therefore x=a/b-->x=3/1--->x=3

roster form of B={3}

C = { y | y is the remainder of x/5, x ∈ ℤ⁺ }

y is remainder of x/5 that is, y=x%5

x belongs to z+ , i am not considering zero as positive integer , so x starts from 1,2,3,.....

if x=1,y=x%5--> y=1%5--->y=1

if x=2,y=x%5--> y=2%5--->y=2

if x=3,y=x%5--> y=3%5--->y=3

if x=4,y=x%5--> y=4%5--->y=4

if x=5,y=x%5--> y=5%5--->y=0

if x=6,y=x%5--> y=6%5--->y=1

...

therefore the remainders are 0,1,2,3,4,0,1,... are repeated

so, roster form of c={0,1,2,3,4}.