Exercise 3.1 — Class 9 Mathematics Solutions

Solution

1. Identify the pattern of the given elements.

   Let the given set be $A = \{1, 4, 9, 16, 25, 36, \dots, 484\}$. Observe that each element is the square of a consecutive natural number:

   $$\begin{aligned} 1 &= 1^2 \\ 4 &= 2^2 \\ 9 &= 3^2 \\ 16 &= 4^2 \\ &\;\;\vdots \\ 484 &= 22^2 \end{aligned}$$

2. Define the rule and condition in set-builder notation.

   Each element $x$ can be expressed as $x = n^2$, where $n \in \mathbb{N}$ and $1 \le n \le 22$ (or $n \le 22$).

3. Write the final set-builder form.

   $$\{x \mid x = n^2, n \in \mathbb{N} \land 1 \le n \le 22\}$$

Solution

1. Identify the pattern of the given elements.

   Let the given set be $B = \{2, 4, 8, 16, \dots, 256\}$. Observe that each element is an increasing positive power of $2$:

   $$\begin{aligned} 2 &= 2^1 \\ 4 &= 2^2 \\ 8 &= 2^3 \\ 16 &= 2^4 \\ &\;\;\vdots \\ 256 &= 2^8 \end{aligned}$$

2. Define the rule and condition.

   Each element $x$ is of the form $x = 2^n$, where $n$ is a natural number such that $1 \le n \le 8$.

3. Write the final set-builder form.

   $$\{x \mid x = 2^n, n \in \mathbb{N} \land 1 \le n \le 8\}$$

   (Note: If the textbook series terminates at $1024 = 2^{10}$, the condition is $1 \le n \le 10$.)

Solution

1. Identify the set of numbers.

   The elements $0, \pm 1, \pm 2, \dots, \pm 1000$ are integers ranging from $-1000$ up to $+1000$ inclusive.

2. Formulate the set-builder rule.

   • The variable $x$ belongs to the set of integers $\mathbb{Z}$.
   • The lower and upper bounds are $-1000 \le x \le 1000$.

3. Write the set-builder form.

   $$\{x \mid x \in \mathbb{Z} \land -1000 \le x \le 1000\}$$

Solution

1. Identify the pattern of the given elements.

   Let the given set be $C = \{6, 12, 18, \dots, 120\}$. Notice that all elements are consecutive positive multiples of $6$:

   $$\begin{aligned} 6 &= 6 \times 1 \\ 12 &= 6 \times 2 \\ 18 &= 6 \times 3 \\ &\;\;\vdots \\ 120 &= 6 \times 20 \end{aligned}$$

2. Define the rule and condition.

   Each element $x$ can be written as $x = 6n$, where $n \in \mathbb{N}$ and $1 \le n \le 20$.

3. Write the set-builder form.

   $$\{x \mid x = 6n, n \in \mathbb{N} \land 1 \le n \le 20\}$$

Solution

1. Identify the pattern of the given elements.

   The elements $100, 102, \dots, 400$ are consecutive even natural numbers starting at $100$ and ending at $400$.

2. Formulate using standard even numbers notation $E$.

   • $x \in E$ (where $E$ denotes the set of even natural numbers).
   • The range is $100 \le x \le 400$.

3. Write the set-builder form.

   $$\{x \mid x \in E \land 100 \le x \le 400\}$$

Solution

1. Identify the pattern of the given elements.

   Let the given set be $\{1, 3, 9, 27, 81, \dots\}$. Observe that each element is a non-negative integral power of $3$:

   $$\begin{aligned} 1 &= 3^0 \\ 3 &= 3^1 \\ 9 &= 3^2 \\ 27 &= 3^3 \\ 81 &= 3^4 \\ &\;\;\vdots \end{aligned}$$

2. Formulate the set-builder rule.

   Each element $x$ can be written as $x = 3^n$, where $n$ belongs to the set of whole numbers $W = \{0, 1, 2, 3, \dots\}$.

3. Write the set-builder form.

   $$\{x \mid x = 3^n, n \in W\}$$

Solution

1. Analyze the given elements.

   Let the given set be $\{1, 2, 4, 5, 10, 20, 25, 50, 100\}$. Notice that every number in this set divides $100$ completely without leaving a remainder:

   $$\begin{aligned} 100 &= 1 \times 100 \\ 100 &= 2 \times 50 \\ 100 &= 4 \times 25 \\ 100 &= 5 \times 20 \\ 100 &= 10 \times 10 \end{aligned}$$

2. Formulate the rule in words and notation.

   These are all the positive factors (or divisors) of $100$.

3. Write the set-builder form.

   $$\{x \mid x \text{ is a positive divisor of } 100\}$$

Solution

1. Identify the pattern of the given elements.

   Let the given set be $\{5, 10, 15, \dots, 100\}$. Notice that all elements are consecutive positive multiples of $5$:

   $$\begin{aligned} 5 &= 5 \times 1 \\ 10 &= 5 \times 2 \\ 15 &= 5 \times 3 \\ &\;\;\vdots \\ 100 &= 5 \times 20 \end{aligned}$$

2. Define the rule and condition.

   Each element $x$ can be written as $x = 5n$, where $n \in \mathbb{N}$ and $1 \le n \le 20$.

3. Write the set-builder form.

   $$\{x \mid x = 5n, n \in \mathbb{N} \land 1 \le n \le 20\}$$

Solution

1. Interpret the verbal description.

   We are given: "The set of all integers between $-100$ and $1000$".

   • The elements belong to the set of integers $\mathbb{Z}$.
   • The word "between" signifies strict inequality, excluding the two boundary values $-100$ and $1000$.

2. Formulate the set-builder notation.

   $$\{x \mid x \in \mathbb{Z} \land -100 < x < 1000\}$$

Solution

1. Understand the condition.

   We need to find all positive multiples of $3$ that are less than or equal to $36$.

2. List the multiples systematically.

   $$\begin{aligned} 3 \times 1 &= 3 \\ 3 \times 2 &= 6 \\ 3 \times 3 &= 9 \\ 3 \times 4 &= 12 \\ 3 \times 5 &= 15 \\ 3 \times 6 &= 18 \\ 3 \times 7 &= 21 \\ 3 \times 8 &= 24 \\ 3 \times 9 &= 27 \\ 3 \times 10 &= 30 \\ 3 \times 11 &= 33 \\ 3 \times 12 &= 36 \end{aligned}$$

3. Write in tabular form.

   $$\{3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36\}$$

Solution

1. State the given algebraic condition.

   We are given $x \in \mathbb{R}$ satisfying the linear equation:

   $$2x + 1 = 0$$

2. Solve the linear equation for $x$.

   $$\begin{aligned} 2x + 1 &= 0 \\ 2x &= -1 \\ x &= -\frac{1}{2} \end{aligned}$$

3. Verify membership in the domain.

   Since $-\frac{1}{2}$ is a real number ($-\frac{1}{2} \in \mathbb{R}$), it is a valid element of the set.

4. Write in tabular form.

   $$\left\{-\frac{1}{2}\right\}$$

Solution

1. Recall the definition of the prime numbers set $P$.

   Prime numbers are natural numbers greater than $1$ that have exactly two distinct positive divisors ($1$ and itself):

   $$P = \{2, 3, 5, 7, 11, 13, 17, 19, \dots\}$$

2. Apply the condition $x < 12$.

   The prime numbers strictly less than $12$ are:

   $$2, 3, 5, 7, 11$$

3. Write in tabular form.

   $$\{2, 3, 5, 7, 11\}$$

Solution

1. Find the prime factorization of $128$.

   $$128 = 2^7$$

2. List all positive divisors systematically as powers of $2$.

   $$\begin{aligned} 2^0 &= 1 \\ 2^1 &= 2 \\ 2^2 &= 4 \\ 2^3 &= 8 \\ 2^4 &= 16 \\ 2^5 &= 32 \\ 2^6 &= 64 \\ 2^7 &= 128 \end{aligned}$$

3. Write in tabular form.

   $$\{1, 2, 4, 8, 16, 32, 64, 128\}$$

Solution

1. Identify the allowed values for $n$.

   Given $n \in \mathbb{N}$ and $n < 8$, the possible values of $n$ are:

   $$n \in \{1, 2, 3, 4, 5, 6, 7\}$$

2. Calculate $x = 2^n$ for each value of $n$.

   $$\begin{aligned} n = 1 &\implies x = 2^1 = 2 \\ n = 2 &\implies x = 2^2 = 4 \\ n = 3 &\implies x = 2^3 = 8 \\ n = 4 &\implies x = 2^4 = 16 \\ n = 5 &\implies x = 2^5 = 32 \\ n = 6 &\implies x = 2^6 = 64 \\ n = 7 &\implies x = 2^7 = 128 \end{aligned}$$

3. Write in tabular form.

   $$\{2, 4, 8, 16, 32, 64, 128\}$$

Solution

1. Solve the given equation.

   $$\begin{aligned} x + 4 &= 0 \\ x &= -4 \end{aligned}$$

2. Check if the solution belongs to the natural numbers $\mathbb{N}$.

   The natural numbers are $\mathbb{N} = \{1, 2, 3, 4, \dots\}$. Since $-4$ is a negative integer, $-4 \notin \mathbb{N}$.

3. Conclude tabular form.

   There is no natural number satisfying the given condition. Hence, the set is empty:

   $$\{\ \} \quad \text{or} \quad \emptyset$$

Solution

1. Analyze the condition.

   The condition $x = x$ is the reflexive property of equality, which is identically true for every number in any universal set.

2. Identify all matching elements.

   Since every natural number $x \in \mathbb{N}$ satisfies $x = x$, the set contains all natural numbers:

   $$\mathbb{N} = \{1, 2, 3, 4, 5, \dots\}$$

3. Write in tabular form.

   $$\{1, 2, 3, 4, 5, \dots\}$$

Solution

1. Solve the given equation for $x$.

   $$\begin{aligned} 3x + 1 &= 0 \\ 3x &= -1 \\ x &= -\frac{1}{3} \end{aligned}$$

2. Check if the solution belongs to the set of integers $\mathbb{Z}$.

   The integers are $\mathbb{Z} = \{0, \pm 1, \pm 2, \pm 3, \dots\}$. Since $-\frac{1}{3}$ is a non-integer rational fraction, $-\frac{1}{3} \notin \mathbb{Z}$.

3. Conclude tabular form.

   No integer satisfies the condition. Therefore, the set is empty:

   $$\{\ \} \quad \text{or} \quad \emptyset$$

Solution

1. Recall the definition of a proper subset.

   A set $A$ is a proper subset of $B$ (denoted $A \subset B$) if every element of $A$ is in $B$ and $A \neq B$.

2. List potential proper subsets.

   For the $3$-element set $\{a, b, c\}$, the proper subsets include:

   $$\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{b, c\}, \{a, c\}$$

3. Choose two proper subsets.

   Two proper subsets are:

   $$\{a\} \quad \text{and} \quad \{b\}$$

Solution

1. Identify the proper subsets of $\{0, 1\}$.

   The subsets of $\{0, 1\}$ are $\emptyset, \{0\}, \{1\}, \{0, 1\}$. The proper subsets are all subsets except the set itself:

   $$\emptyset, \{0\}, \{1\}$$

2. Select two proper subsets.

   Two proper subsets are:

   $$\{0\} \quad \text{and} \quad \{1\}$$

Solution

1. Understand the set of natural numbers $\mathbb{N}$.

   $$\mathbb{N} = \{1, 2, 3, 4, 5, 6, \dots\}$$

2. Identify well-defined proper subsets of $\mathbb{N}$.

   • The set of even natural numbers:

   $$E = \{2, 4, 6, 8, \dots\}$$

   • The set of odd natural numbers:

   $$O = \{1, 3, 5, 7, \dots\}$$

   Both $E \subset \mathbb{N}$ and $O \subset \mathbb{N}$ are strictly contained in $\mathbb{N}$.

3. Write the two proper subsets.

   $$\{2, 4, 6, \dots\} \quad \text{and} \quad \{1, 3, 5, \dots\}$$

Solution

1. Understand the set of integers $\mathbb{Z}$.

   $$\mathbb{Z} = \{0, \pm 1, \pm 2, \pm 3, \dots\}$$

2. Identify standard proper subsets of $\mathbb{Z}$.

   • The set of positive integers:

   $$\mathbb{Z}^+ = \{1, 2, 3, \dots\}$$

   • The set of negative integers:

   $$\mathbb{Z}^- = \{-1, -2, -3, \dots\}$$

   Both $\mathbb{Z}^+ \subset \mathbb{Z}$ and $\mathbb{Z}^- \subset \mathbb{Z}$ are valid proper subsets.

3. Write the two proper subsets.

   $$\mathbb{Z}^+ = \{1, 2, 3, \dots\} \quad \text{and} \quad \mathbb{Z}^- = \{-1, -2, -3, \dots\}$$

Solution

1. Understand the set of rational numbers $\mathbb{Q}$.

   Rational numbers include all numbers that can be written in the form $\frac{p}{q}$ where $p, q \in \mathbb{Z}$ and $q \neq 0$.

2. Identify proper subsets.

   Every integer and every natural number is a rational number (with denominator $1$), but $\mathbb{Q}$ contains non-integers (such as $\frac{1}{2}, \frac{3}{4}$). Therefore:

   $$\mathbb{N} \subset \mathbb{Q} \quad \text{and} \quad \mathbb{Z} \subset \mathbb{Q}$$

3. Write the two proper subsets.

   $$\mathbb{N} \quad \text{and} \quad \mathbb{Z}$$

Solution

1. Understand the set of real numbers $\mathbb{R}$.

   The real numbers $\mathbb{R}$ is the union of rational numbers $\mathbb{Q}$ and irrational numbers $\mathbb{Q}'$:

   $$\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}'$$

2. Identify proper subsets.

   Both $\mathbb{Q}$ and $\mathbb{Q}'$ are non-empty, strictly contained subsets of $\mathbb{R}$:

   $$\mathbb{Q} \subset \mathbb{R} \quad \text{and} \quad \mathbb{Q}' \subset \mathbb{R}$$

3. Write the two proper subsets.

   $$\mathbb{Q} \quad \text{and} \quad \mathbb{Q}'$$

Solution

1. Understand the given set.

   Let $S = \{x \mid x \in \mathbb{Q} \land 0 < x \le 2\}$. This set contains all rational numbers strictly between $0$ and $2$, including $2$.

2. Select specific elements in this interval.

   Values such as $\frac{1}{2}$, $\frac{2}{3}$, $1$, and $2$ satisfy $0 < x \le 2$.

3. Form two proper subsets.

   Two proper singleton subsets are:

   $$\left\{\frac{1}{2}\right\} \quad \text{and} \quad \left\{\frac{2}{3}\right\}$$

   (Alternatively: $\{1\}$ and $\{1, 2\}$.)

Solution

1. Yes, the empty set $\emptyset$ (or $\{\ \}$).

   Reason: $\emptyset$ has only one subset (itself), which is improper. Hence, $\emptyset$ has no proper subset.

Solution

1. 1. $\{a, b\}$ contains 2 elements: $a$ and $b$ (Cardinality 2).
   2. $\{\{a, b\}\}$ contains 1 element: the set $\{a, b\}$ itself (Cardinality 1, a singleton set).

Solution

1. Identify the cardinality of the given set.

   The given set is the empty set $\{\ \} = \emptyset$. The number of elements is:

   $$n = 0$$

2. Apply the power set cardinality formula.

   The number of elements in the power set $P(S)$ of a set with $n$ elements is given by:

   $$n(P(S)) = 2^n$$

3. Calculate the result.

   $$\begin{aligned} n(P(S)) &= 2^0 \\ &= 1 \end{aligned}$$

Solution

1. Identify the cardinality of the given set.

   The given set is $\{0, 1\}$. The number of elements is:

   $$n = 2$$

2. Apply the power set cardinality formula.

   $$\begin{aligned} n(P(S)) &= 2^n \\ &= 2^2 \\ &= 4 \end{aligned}$$

Solution

1. Identify the cardinality of the given set.

   The given set contains $7$ distinct elements:

   $$n = 7$$

2. Apply the power set cardinality formula.

   $$\begin{aligned} n(P(S)) &= 2^n \\ &= 2^7 \\ &= 128 \end{aligned}$$

Solution

1. Identify the cardinality of the given set.

   The given set contains $8$ distinct elements (from $0$ to $7$):

   $$n = 8$$

2. Apply the power set cardinality formula.

   $$\begin{aligned} n(P(S)) &= 2^n \\ &= 2^8 \\ &= 256 \end{aligned}$$

Solution

1. Identify the elements and cardinality of the set.

   The given set is $S = \{a, \{b, c\}\}$. It contains exactly two elements:

   1. The element $a$
   2. The set $\{b, c\}$

   Therefore:

   $$n = 2$$

2. Apply the power set cardinality formula.

   $$\begin{aligned} n(P(S)) &= 2^n \\ &= 2^2 \\ &= 4 \end{aligned}$$

Solution

1. Identify the elements and cardinality of the set.

   The given set contains three elements (each being a 2-element set):

   1. $\{a, b\}$
   2. $\{b, c\}$
   3. $\{d, e\}$

   Therefore:

   $$n = 3$$

2. Apply the power set cardinality formula.

   $$\begin{aligned} n(P(S)) &= 2^n \\ &= 2^3 \\ &= 8 \end{aligned}$$

Solution

1. Determine the cardinality and number of subsets.

   Let $A = \{9, 11\}$. Number of elements: $n = 2$. Total number of subsets:

   $$2^n = 2^2 = 4$$

2. List all subsets systematically by cardinality.

   • 0 elements (empty set): $\emptyset$
   • 1 element (singletons): $\{9\}, \{11\}$
   • 2 elements (improper subset): $\{9, 11\}$

3. Write the power set $P(A)$.

   $$P(A) = \{\emptyset, \{9\}, \{11\}, \{9, 11\}\}$$