Sets>>>>>>?

Question

ok
{ x | x^2 =2}
{ x | x^2 =3 and x is an element of Q}
{ x | 2 is less than or equal to x and x is less than or equal to 4 and x^2 is an element of Z}
{ x | x^2 =4 and x is an element of N}

any help in listing the elements of those sets would be fabulous.

are squareroots apart of the set of integers? and for the third set is 3 a number in the set?

Answer 1

{ x | x^2 =2} = {sqrt(2), -sqrt(2)}
{ x | x^2 =3 and x is an element of Q} = the empty set, if x^2=3, then x is irrational.

{ x | 2 is less than or equal to x and x is less than or equal to 4 and x^2 is an element of Z}:

2<=x<=4 and x^2 is a whole number then x^2 is a whole number between 4 and 16

The set is {2, sqrt(5), sqrt(6), sqrt(7),sqrt(8),3, sqrt(10, sqrt(11), sqrt(12), sqrt(13), sqrt(14), sqrt(15), 4}

{ x | x^2 =4 and x is an element of N}

x^2=4 implies that x=2 or x=-2
{x | x^2 = 4} = {-2, 2} but only 2 belongs to N, so
{ x | x^2 =4 and x is an element of N} = {2}

Answer 2

{ x | x^2 =2} =======> x / x² = 2
x = 2 * x²
2 * x² - x = 0

Delta = 1 - 4 *2 * 0 = 1

bascara:
x1 = [-(-1) + 1]/(2*2) = 1/2
x2 = [-(-1) - 1]/(2*2) = 0

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
{ x | x^2 =3 and x is an element of Q}

x / x² = 3
x = 3 * x²
3 * x² - x = 0

Delta = 1 - 4 *3 * 0 = 1

bascara:
x1 = [-(-1) + 1]/(2*3) = 1/3
x2 = [-(-1) - 1]/(2*3) = 0

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

{ x | 2 is less than or equal to x and x is less than or equal to 4 and x^2 is an element of Z}

x/2 < x
x < 4 ==> x = {..., -3, -2, -1, 0, 1, 2, 3, 4} [Z]
then x = {0, 2, 4}


xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

{ x | x^2 =4 and x is an element of N}

x / x² = 4
x = 4 * x²
4 * x² - x = 0

Delta = 1 - 4 *4 * 0 = 1

bascara:
x1 = [-(-1) + 1]/(2*4) = 1/4 {not an element of N}
x2 = [-(-1) - 1]/(2*4) = 0