1. Write the following sets in set builder notation:
(i) \(\{1, 4, 9, 16, 25, 36, \dots, 484\}\)
\[\{x \, | \, x = n^2, \, n \in \mathbb{N}, \, 1 \leq n \leq 22\}\]
(ii) \(\{2, 4, 8, 16, 32, 64, \dots, 150\}\)
\[\{x \, | \, x = 2^n, \, n \in \mathbb{N}, \, 1 \leq n \leq 7\}\]
(iii) \(\{0, \pm 1, \pm 2, \dots, \pm 1000\}\)
\[\{x \, | \, x \in \mathbb{Z}, \, -1000 \leq x \leq 1000\}\]
(iv) \(\{6, 12, 18, \dots, 120\}\)
\[\{x \, | \, x = 6n, \, n \in \mathbb{N}, \, 1 \leq n \leq 20\}\]
(v) \(\{100, 102, 104, \dots, 400\}\)
\[\{x \, | \, x = 100 + 2n, \, n \in \mathbb{N}, \, 0 \leq n \leq 150\}\]
(vi) \(\{1, 3, 9, 27, 81, \dots\}\)
\[\{x \, | \, x = 3^n, \, n \in \mathbb{N}\}\]
(vii) \(\{1, 2, 4, 5, 10, 20, 25, 50, 100\}\)
\[\{x \, | \, x \text{ is a divisor of } 100\}\]
(viii) \(\{5, 10, 15, \dots, 100\}\)
\[\{x \, | \, x = 5n, \, n \in \mathbb{N}, \, 1 \leq n \leq 20\}\]
(ix) The set of all integers between \(-100\) and \(1000\):
\[\{x \, | \, x \in \mathbb{Z}, \, -100 \leq x \leq 1000\}\]
2. Write each of the following sets in tabular form:
(i) \(\{x \, | \, x \text{ is a multiple of } 3 \, \land \, x \leq 35\}\)
\[\{3, 6, 9, 12, 15, 18, 21, 24, 27, 30\}\]
(ii) \(\{x \, | \, x \in \mathbb{R} \, \land \, 2x + 1 = 0\}\)
\[\{-\frac{1}{2}\}\]
(iii) \(\{x \, | \, x \in \mathbb{P} \, \land \, x < 12\}\)
\[\{2, 3, 5, 7, 11\}\]
(iv) \(\{x \, | \, x \text{ is a divisor of } 128\}\)
\[\{1, 2, 4, 8, 16, 32, 64, 128\}\]
(v) \(\{x \, | \, x = 2^n, \, n \in \mathbb{N}, \, n < 8\}\)
\[\{2, 4, 8, 16, 32, 64, 128\}\]
(vi) \(\{x \, | \, x \in \mathbb{N} \, \land \, x + 4 = 0\}\)
\[\{\}\]
(vii) \(\{x \, | \, x \in \mathbb{N} \, \land \, x = x\}\)
\[\{1, 2, 3, 4, 5, \dots\}\]
(viii) \(\{x \, | \, x \in \mathbb{Z} \, \land \, 3x + 1 = 0\}\)
\[\{-\frac{1}{3}\}\]
3. Write two proper subsets of each of the following sets:
(i) \(\{a, b, c\}\)
\[\{a\}, \{b, c\}\]
(ii) \(\{0, 1\}\)
\[\{\}, \{1\}\]
(iii) \(\mathbb{N}\)
\[\{1, 2\}, \{3, 4\}\]
(iv) \(\mathbb{Z}\)
\[\{-1, 0\}, \{2, 3\}\]
(v) \(Q\)
\[\{\frac{1}{2}\}, \{1, 2\}\]
(vi) \(R\)
\[\{\pi\}, \{0.1, 2.5\}\]
(vii) \(\{x \, | \, x \in \mathbb{Q} \, \land \, 0 < x \leq 2\}\)
\[\{1\}, \{\frac{1}{2}, \frac{3}{2}\}\]
4. Is there any set which has no proper subset? If so, name that set.
\[\{\}\]
5. What is the difference between \(\{a, b\}\) and \(\{\{a, b\}\}\)?
\[\{a, b\} \text{ contains the elements } a, b\]
\[\{\{a, b\}\} \text{ contains the single element } \{a, b\}\]
6. What is the number of elements of the power set of each of the following sets?
(i) \(\{\}\)
\[2^0 = 1\]
(ii) \(\{0, 1\}\)
\[2^2 = 4\]
(iii) \(\{1, 2, 3, 4, 5, 6, 7\}\)
\[2^7 = 128\]
(iv) \(\{a, b, c\}\)
\[2^3 = 8\]
(v) \(\{\{a, b\}, \{b, c\}, \{d, e\}\}\)
\[2^3 = 8\]
7. Write down the power set of each of the following sets:
(i) \(\{9, 11\}\)
\[\{\{\}, \{9\}, \{11\}, \{9, 11\}\}\]
(ii) \(\{+, -, \times, \div\}\)
\[\{\{\}, \{+\}, \{-\}, \{\times\}, \{\div\}, \{+, -\}, \{+, \times\}, \{\times, \div\}, \dots, \{+, -, \times, \div\}\}\]
(iii) \(\{\phi\}\)
\[\{\{\}, \{\phi\}\}\]
(iv) \(\{a, \{b, c\}\}\)
\[\{\{\}, \{a\}, \{\{b, c\}\}, \{a, \{b, c\}\}\}\]
