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:

   $$\begin{array}{cc}1& =1^2\\ 4& =2^2\\ 9& =3^2\\ 16& =4^2\\ & ⋮\\ 484& ={22}^2\end{array}$$

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.

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

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 $2$:

   $$\begin{array}{cc}2& =2^1\\ 4& =2^2\\ 8& =2^3\\ 16& =2^4\\ & ⋮\\ 256& =2^8\end{array}$$

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.

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

   (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.

   $$\{x|x\in \mathbb{Z}\wedge -1000\le x\le 1000\}$$

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 $6$:

   $$\begin{array}{cc}6& =6\times 1\\ 12& =6\times 2\\ 18& =6\times 3\\ & ⋮\\ 120& =6\times 20\end{array}$$

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.

   $$\{x|x=6n,n\in \mathbb{N}\wedge 1\le n\le 20\}$$

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.

   $$\{x|x\in E\wedge 100\le x\le 400\}$$

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 $3$:

   $$\begin{array}{cc}1& =3^0\\ 3& =3^1\\ 9& =3^2\\ 27& =3^3\\ 81& =3^4\\ & ⋮\end{array}$$

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.

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

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:

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

2. Formulate the rule in words and notation.

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

3. Write the set-builder form.

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

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 $5$:

   $$\begin{array}{cc}5& =5\times 1\\ 10& =5\times 2\\ 15& =5\times 3\\ & ⋮\\ 100& =5\times 20\end{array}$$

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.

   $$\{x|x=5n,n\in \mathbb{N}\wedge 1\le n\le 20\}$$

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.

   $$\{x|x\in \mathbb{Z}\wedge -100<x<1000\}$$

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.

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

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 $x\in ℝ$ satisfying the linear equation:

   $$2x+1=0$$

2. Solve the linear equation for $x$.

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

3. Verify membership in the domain.

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

4. Write in tabular form.

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

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 $2$.

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

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 $n$.

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

3. Write in tabular form.

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

Solution

1. Solve the given equation.

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

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:

   $$\{\}\text{or}\varnothing$$

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 $x$.

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

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:

   $$\{\}\text{or}\varnothing$$

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 $A\ne B$.

2. List potential proper subsets.

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

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

3. Choose two proper subsets.

   Two proper subsets are:

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

Solution

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

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

   $$\varnothing ,\{0\},\{1\}$$

2. Select two proper subsets.

   Two proper subsets are:

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

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.

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

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.

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

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 $q\ne 0$.

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:

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

3. Write the two proper subsets.

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

Solution

1. Understand the set of real numbers $ℝ$.

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

   $$ℝ=\mathbb{Q}\cup {\mathbb{Q}}^{\prime }$$

2. Identify proper subsets.

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

   $$\mathbb{Q}\subset ℝ\text{and}{\mathbb{Q}}^{\prime }\subset ℝ$$

3. Write the two proper subsets.

   $$\mathbb{Q}\text{and}{\mathbb{Q}}^{\prime }$$

Solution

1. Understand the given set.

   Let $S=\{x|x\in \mathbb{Q}\wedge 0<x\le 2\}$. 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:

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

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

Solution

Solution

Solution

1. Identify the cardinality of the given set.

   The given set is the empty set $\{\}=\varnothing$. 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.

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

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.

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

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.

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

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.

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

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.

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

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.

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

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): $\varnothing$
   • 1 element (singletons): $\{9\},\{11\}$
   • 2 elements (improper subset): $\{9,11\}$

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

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

Solution

1. Determine the cardinality and number of subsets.

   Let $B=\{+,-,\times ,÷\}$. Number of elements: $n=4$. Total number of subsets:

   $$2^n=2^4=16$$

2. List all 16 subsets systematically by number of elements.

   • 0 elements (1 subset): $\varnothing$
   • 1 element (4 subsets): $\{+\},\{-\},\{\times \},\{÷\}$
   • 2 elements (6 subsets): $\{+,-\},\{+,\times \},\{+,÷\},\{-,\times \},\{-,÷\},\{\times ,÷\}$
   • 3 elements (4 subsets): $\{+,-,\times \},\{+,-,÷\},\{+,\times ,÷\},\{-,\times ,÷\}$
   • 4 elements (1 subset): $\{+,-,\times ,÷\}$

3. Write the complete power set $P(B)$.

   $$\begin{array}{cc}P(B)=\{& \varnothing ,\\ & \{+\},\{-\},\{\times \},\{÷\},\\ & \{+,-\},\{+,\times \},\{+,÷\},\{-,\times \},\{-,÷\},\{\times ,÷\},\\ & \{+,-,\times \},\{+,-,÷\},\{+,\times ,÷\},\{-,\times ,÷\},\\ & \{+,-,\times ,÷\}\}\end{array}$$

Solution

1. Determine the cardinality of the given set.

   Let $C=\{\varnothing \}$. Notice that $C$ is a singleton set containing the empty set $\varnothing$ as its sole element. Number of elements: $n=1$. Total number of subsets:

   $$2^n=2^1=2$$

2. List all subsets.

   • 0 elements (empty set): $\varnothing$
   • 1 element (the set itself): $\{\varnothing \}$

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

   $$P(C)=\{\varnothing ,\{\varnothing \}\}$$

Solution

1. Determine the cardinality and number of subsets.

   Let $D=\{a,\{b,c\}\}$. The set contains $2$ elements: the element $a$ and the set $\{b,c\}$. Number of elements: $n=2$. Total number of subsets:

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

2. List all subsets.

   • 0 elements: $\varnothing$
   • 1 element: $\{a\},\{\{b,c\}\}$
   • 2 elements: $\{a,\{b,c\}\}$

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

   $$P(D)=\{\varnothing ,\{a\},\{\{b,c\}\},\{a,\{b,c\}\}\}$$