Set Builder Form
We can also define a set by stating properties which its elements must satisfy. For example, let E denote the set of all even natural numbers, then we use a letter, usually x, to represent an arbitrary element and we write:Illustration
Let H be the set of all human beings on the earth, that is = {x / x is a human being on the earth}.
Illustration
Let l P be a fixed line in the plane. Let P = {x / x is a line in the plane which is perpendicular to l}.
Illustration
Write H = {Fluorine, Chlorine, Bromine, Iodine, Astatine} in set - builder form.
Solution
H ={x : x is a halogen}
Example 1
Match each of the following sets in the column A described in roster form with the same set in the column B described in the set-builder form.
Column A |
Column B |
(a) {M, A, T, H, E, I, C, S} | (a) {x : x âˆˆ N and x is a multiple of 5} |
(b) {5, 10, 15, 20, 25, ...} | (b) {x : x âˆˆ R, x â‰ x} |
(c) {} | (c) {x âˆ x is a letter of the word FOLLOW} |
(d) {F, L, O, W} | (d) {x âˆ x is a letter of the word 'MATHEMATICS'} |
(e) {1, 2, 3, 4,... 100} | (e) |
(f) | (f) {x : x âˆˆ N and x â‰¤ 100} |
Solution
(a) â†” (d)
(b) â†” (a)
(c) â†” (b)
(d) â†” (c)
(e) â†” (f)
(f) â†” (e)
Example 2
Write the following sets in set-builder form.
- A = {1, 4, 9, 16, 25, ...}
- B = {a, e, i, o, u}
- C = {2, 4, 8, 16, 32, 64}
- {2, 3, 5, 7, 11, 13, 17, 19, 23, ...}
- {1, 2, 3, 4, 6, 12, 24, 48}
- A = {x^{2} âˆ x âˆˆ N}
- B = {x : x is a vowel of the English alphabet}
- C = {x âˆ x > 1 and x is a divisor of 64}
- {x âˆ x is a prime number}
- {x : x is a divisor of 48}.