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Question: Write the following sets in the set builder form : (i) O = {0} (ii) P = {2} (iii) Q = {1, 4, 9, 16} ...

Write the following sets in the set builder form : (i) O = {0} (ii) P = {2} (iii) Q = {1, 4, 9, 16} (iv) R = {2, 3, 5, 7} (v) S = {1, 3, 9, 27} (vi) T = {A, E, I, O, U} (vii) V = {0, 3, 6, 9, ...}

Answer

(i) O={x∣x=0}O = \{x \mid x = 0\}

(ii) P={x∣x=2}P = \{x \mid x = 2\}

(iii) Q={x∣x=n2,n∈N and 1≤n≤4}Q = \{x \mid x = n^2, n \in \mathbb{N} \text{ and } 1 \le n \le 4\}

(iv) R={x∣x is a prime number and x<10}R = \{x \mid x \text{ is a prime number and } x < 10\}

(v) S={x∣x=3n,n∈W and 0≤n≤3}S = \{x \mid x = 3^n, n \in \mathbb{W} \text{ and } 0 \le n \le 3\}

(vi) T={x∣x is a vowel in the English alphabet}T = \{x \mid x \text{ is a vowel in the English alphabet}\}

(vii) V={x∣x=3n,n∈W}V = \{x \mid x = 3n, n \in \mathbb{W}\}

Explanation

Solution

To write a set in set-builder form, we describe the common property shared by all the elements of the set.

(i) O = {0}

This set contains only the element 0.
In set-builder form:
O={x∣x=0}O = \{x \mid x = 0\}

(ii) P = {2}

This set contains only the element 2.
In set-builder form:
P={x∣x=2}P = \{x \mid x = 2\}

(iii) Q = {1, 4, 9, 16}

Observe the pattern:
1=121 = 1^2
4=224 = 2^2
9=329 = 3^2
16=4216 = 4^2
The elements are squares of natural numbers from 1 to 4.
In set-builder form:
Q={x∣x=n2,n∈N and 1≤n≤4}Q = \{x \mid x = n^2, n \in \mathbb{N} \text{ and } 1 \le n \le 4\}

(iv) R = {2, 3, 5, 7}

Observe the pattern:
These are the first four prime numbers.
In set-builder form:
R={x∣x is a prime number and x<10}R = \{x \mid x \text{ is a prime number and } x < 10\} (or x≤7x \le 7)

(v) S = {1, 3, 9, 27}

Observe the pattern:
1=301 = 3^0
3=313 = 3^1
9=329 = 3^2
27=3327 = 3^3
The elements are powers of 3, where the exponent ranges from 0 to 3.
In set-builder form:
S={x∣x=3n,n∈W and 0≤n≤3}S = \{x \mid x = 3^n, n \in \mathbb{W} \text{ and } 0 \le n \le 3\} (where W\mathbb{W} denotes whole numbers)
Alternatively, using integers: S={x∣x=3n,n∈Z and 0≤n≤3}S = \{x \mid x = 3^n, n \in \mathbb{Z} \text{ and } 0 \le n \le 3\}

(vi) T = {A, E, I, O, U}

Observe the pattern:
These are the vowels in the English alphabet.
In set-builder form:
T={x∣x is a vowel in the English alphabet}T = \{x \mid x \text{ is a vowel in the English alphabet}\}

(vii) V = {0, 3, 6, 9, ...}

Observe the pattern:
0=3×00 = 3 \times 0
3=3×13 = 3 \times 1
6=3×26 = 3 \times 2
9=3×39 = 3 \times 3
...
The elements are non-negative multiples of 3.
In set-builder form:
V={x∣x=3n,n∈W}V = \{x \mid x = 3n, n \in \mathbb{W}\} (where W\mathbb{W} denotes whole numbers)
Alternatively, using integers: V={x∣x=3n,n∈Z and n≥0}V = \{x \mid x = 3n, n \in \mathbb{Z} \text{ and } n \ge 0\}