Class 9 Mathematics | Punjab Curriculum and Textbook Board Syllabus 2025
A collection of well-defined distinct object is called set. It is denoted by capital letters $A$, $B$, $C$ etc. For example, $A = \left\{ 1,2,3,4 \right\}$
A set can be described using three different methods:
A set is described in words without listing its elements. For example, the set of all vowels in the English alphabet.
A set is described by listing its elements within curly brackets $\left\{ \right\}$. For example, if A is the set of vowels, we write:
$$A = \left\{ a,e,i,o,u \right\}$$
A set is described by stating a property that all its elements share. For example, the set of vowels can be written as:
$$A = \left\{ x|x\ is\ a\ vowel\ of\ the\ English\ alphabet \right\}$$
Note: In algebra, we usually deal with sets of numbers. Such sets, along with their names, are given below:
A set with only one element is called a singleton set. For example, $\{ 3\}$, $\{ a\}$, and $\{ Saturday\}$ are singleton sets.
The set with no elements (zero number of elements) is called an empty set, null set, or void set. The empty set is denoted by the symbol $\varnothing$ or $\left\{ \right\}$.
Two sets $A$ and $B$ are equal if they have exactly the same elements or if every element of set $A$ is an element of set $B$. If two sets $A$ and $B$ are equal, we write $A\ = \ B$.
Thus, the sets $\{ 1,\ 2,\ 3\}$ and $\{ 2,\ 1,\ 3\}$ are equal.
Two sets $A$ and $B$ are equivalent if they have the same number of elements. For example, if $A = \{ a,\ b,\ c,\ d,\ e\}$ and $B = \{ 1,\ 2,\ 3,\ 4,\ 5\}$, then $A$ and $B$ are equivalent sets. The symbol $\sim$ is used to represent equivalent sets. Thus, we can write $A \sim B$.
If every element of a set $A$ is an element of set $B$, then $A$ is a subset of $B$. Symbolically, this is written as $A \subseteq B\ (A\ is\ a\ subset\ of\ B)$.
In such a case, we say B is a superset of A. Symbolically, this is written as: $B \supseteq A\ (B\ is\ a\ superset\ of\ A)$
If $A$ is a subset of $B$ and $B$ contains at least one element that is not an element of $A$, then $A$ is said to be a proper subset of $B$. In such a case, we write:
$$A \subset B\ (A\ is\ a\ proper\ subset\ of\ B)$$
If $A$ is a subset of $B$ and $A = B$, then we say that $A$ is an improper subset of $B$. From this definition, it also follows that every set is a subset of itself and is called an improper subset.
For example, let $A = \{ a,\ b,\ c\},$ $B = \{ c,\ a,\ b\}$ and $C = \{ a,\ b,\ c,\ d\}$, then clearly: $A\ \subset \ C,\ B\ \subset \ C$ but $A\ = \ B$.
Note: Notice that each of sets $A$ and $B$ is an improper subset of the other because $A = B$
The set that contains all objects or elements under consideration is called the universal set or the universe of discourse. It is denoted by $U$.
The power set of a set $S$ denoted by $P(S)$ is the set containing all the possible subsets of $S$. If S is a finite set with $n(S) = m$, representing the number of elements in set$\ S$, then:
$$n\left\{ P(S) \right\} = 2^{m}$$
This represents the number of elements in the power set.
Note:
The union of two sets $A$ and $B$, denoted by $A \cup B$, is the set of all elements that belong to $A$ or $B$.
Symbolically,
$$A \cup B = \left\{ x|x \in A\ \vee \ x \in B \right\}$$
For example, if $A = \left\{ 1,2 \right\}$ and $B = \left\{ 1,3 \right\}$, then
$${A \cup B = \left\{ 1,2 \right\} \cup \left\{ 1,3 \right\}}{A \cup B = \left\{ 1,2,3 \right\}}$$
The intersection of two sets $A$ and $B$, written as $A \cap B$, is the set of all elements that belong to both $A$ and $B$. Symbolically,
$$A \cap B = \left\{ x|x \in A\ \land \ x \in B \right\}$$
For example, if $A = \left\{ 1,2 \right\}$ and $B = \left\{ 1,3 \right\}$, then
$${A \cap B = \left\{ 1,2 \right\} \cap \left\{ 1,3 \right\}}{A \cap B = \left\{ 1 \right\}}$$
The set difference of $A$ and $B$ denoted by $A - B$, consists of all elements that belong to $A$ but do not belong to $B$. Symbolically,
$${A - B = \left\{ x|x \in A\ \land \ x \notin B \right\}}{and\ \ \ \ \ \ \ \ \ B - A = \left\{ x|x \in B\ \land \ x \notin A \right\}}$$
For example, if $A = \left\{ 1,2 \right\}$ and $B = \left\{ 1,3 \right\}$, then
$${A - B = \left\{ 1,2 \right\} - \left\{ 1,3 \right\}}{A - B = \left\{ 2 \right\}}$$
The complement of a set $A$, denoted by $A'$ or $A^{c}$, relative to the universal set $U$ is the set of all elements of $U$ that do not belong to $A$. Symbolically,
$${A' = U - A}{A' = \left\{ x|x \in U\ \land \ x \notin A \right\}}$$
For example, if $U = \left\{ 1,2,3,4,5 \right\}$ and $A = \left\{ 1,3 \right\}$, then
$${A' = U - A}{A' = \left\{ 1,2,3,4,5 \right\} \cup \left\{ 1,3 \right\}}{A' = \left\{ 2,4,5 \right\}}$$
If the intersection of two sets is the empty set, the sets are said to be disjoint. For example, if:
$${{S\ }_{1} = \ The\ set\ of\ odd\ natural\ numbers}{S_{2}\ = \ The\ set\ of\ even\ natural\ numbers}$$
Then, ${S\ }_{1}$ and $S_{2}$ are disjoint sets because they have no common elements. Similarly, the set of arts students and the set of science students in a school are disjoint sets.
If the intersection of two sets is non-empty but neither is a subset of the other, the sets are called overlapping sets. For example: If
$${L\ = \ \left\{ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8,\ 9 \right\}}{M\ = \ \left\{ 5,\ 6,\ 7,\ 8,\ 9,\ 10 \right\}}$$
then $L$ and $M$ are overlapping sets because they have common elements.
British mathematician john Venn (1834-1923) introduced rectangle for a universal set $U$ and its subsets $A$ and $B$ as closed figures inside this rectangle.
The cardinality of a set is defined as the total number of elements of a set. It represents the size of the set.
For a non-empty set, the cardinality of a set is denoted by $n(A)$. For example, if
$$A = \left\{ 1,3,5,7,9,11 \right\}$$
then $n(A) = 6$ because the set has $6$ elements.
To find the cardinality of a set, we use a rule called the Inclusion-Exclusion Principle, which helps calculate the number of elements in the union of two or more sets by avoiding overcounting.
If $A$ and $B$ are finite sets, then:
$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$
Note: $A \cup B$ and $A \cap B$ are also finite sets.
If $A$, $B$, and $C$ are finite sets, then:
$$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$$
Note: $A \cup B \cup C$, $A \cap B$, $A \cap C$, $B \cap C$, and $A \cap B \cap C$ are also finite sets.
In everyday use, a relation refers to an abstract connection between two people or objects, such as: $(Teacher,\ Pupil)$, $(Mother,\ Son)$, $(Husband,\ Wife)$, $(Brother,\ Sister)$, $(Friend,\ Friend)$ etc.
In mathematics, a relation is any set of ordered pairs. The relationship between the components of an ordered pair may or may not be mentioned.
Examples of Mathematical Relations:
Let $A$ and $B$ be two non-empty sets. The Cartesian product is the set of all ordered pairs $(x,\ y)$ such that $x \in A$ and $y \in B$ and is denoted by $A \times B$. Symbolically,
$$A \times B = \left\{ x|x \in A\ \land \ y \in B \right\}$$
Any subset of the Cartesian product $A \times B$ is called a binary relation or simply a relation, from $A$ to $B$. It is usually represented by the letter $r$.
Domain: The domain of a relation is the set of first elements of the ordered pairs. It is written as $Dom\ r$.
Range: The range of a relation is the set of second elements of the ordered pairs. It is written as $Ran\ r$.
Example: if $A = \left\{ 1,2,3 \right\}$ and $B = \left\{ 2,3 \right\}$, then relation $r:A \rightarrow B$ such that $r = \left\{ (x,y)|x < y \right\}$
$${A \times B = \left\{ 1,2,3 \right\} \times \left\{ 2,3 \right\}}{A \times B = \left\{ (1,2),(1,3),\ (2,2),\ (2,3),(3,2),\ (3,3) \right\}}$$
Since $r = \left\{ (x,y)|x < y \right\}$, so
$${r = \left\{ (1,2),(1,3),\ (2,3) \right\}}{Dom\ r = \left\{ 1,2 \right\}}{Ran\ r = \left\{ 2,3 \right\}}$$
If $A$ is a non-empty set, any subset of $A\ \times \ A$ is called a relation on $A$.
A very important particular type of relation is a function defined as below:
Let $A$ and $B$ be two non-empty sets such that:
The function $f$ is also written as: $f:A \rightarrow B$
If $\mathbf{A = \{ 0,1,2,3,4\}}$ and $\mathbf{B = \{ 3,5,7,9,11\}}$ define a function $\mathbf{f:A \rightarrow B}$ where:
$$\mathbf{f = \{(x,y) \mid y = 2x + 3,x \in A,y \in B\}}$$
Find the value of function f, its domain, co-domain, and range.
| $\mathbf{x}$ (Domain $\mathbf{A}$) | $\mathbf{y\ = \ 2x\ + \ 3}$ (Range $\mathbf{B}$) |
|---|---|
| $$\mathbf{0}$$ | $$2(0)\ + \ 3\ = \ 3$$ |
| $$\mathbf{1}$$ | $$2(1)\ + \ 3\ = \ 5$$ |
| $$\mathbf{2}$$ | $$2(2)\ + \ 3\ = \ 7$$ |
| $$\mathbf{3}$$ | $$2(3)\ + \ 3\ = \ 9$$ |
| $$\mathbf{4}$$ | $$2(4)\ + \ 3\ = \ 11$$ |
Thus, the function is:
$$f = \{(0,3),(1,5),(2,7),(3,9),(4,11)\}$$
If $f:A \rightarrow B$ is a function, then $A$ is called the domain of $f$ and $B$ is called co-domain of $f.$
If a function $f:A \rightarrow B$ is such $that\ Range\ f \subset B$ i.e., $Range\ f \neq B$, then $f$ is said to be a function from $A$ into $B$.
In figure, $f$ is clearly a function. But $Range\ f\ \neq \ B$. Therefore, $f$ is an into function from $A$ into $B$.
If a function $f:A \rightarrow B$ is such that second elements of no two of its ordered pairs are the same, then it is called an injective function.
The function shown in figure is such a function.
If a function $f:A \rightarrow B$ is such that $Range\ f\ = \ B$, $i.e.,$ every element of $B$ is the image of some element of A, then $f$ is called an onto function or a surjective function.
A function $f:A \rightarrow B$ is said to be a Bijective function if it is both $one - one$ and $onto$. Such a function is also called a $(1 - 1)$ correspondence between the sets $A$ and $B$.
Example: $(a,\ z)$, $(b,\ x)$, and $(c,\ y)$ are the pairs of corresponding elements.
In this case, $f\ = \ \{(a,\ z),\ (b,\ x),\ (c,\ y)\}$ is a bijective function or $(1 - 1)$ correspondence between the sets $A$ and $B$.
We know that set-builder notation is more suitable for infinite sets. The same applies to a function comprising an infinite number of ordered pairs.
Example: Consider the function:
$$f = \left\{ (-1,1),\ (0,0),\ (1,1),\ (2,4),\ (3,9),\ (4,16),\ \ldots \right\}$$
$$\text{Domain of } f = \left\{ -1,\ 0,\ 1,\ 2,\ 3,\ 4,\ \ldots \right\}$$
$$\text{Range of } f = \left\{ 0,\ 1,\ 4,\ 9,\ 16,\ \ldots \right\}$$
This function may also be written as:
$$f = \left\{ (x, y)\ |\ y = x^2,\ x \in \mathbb{N} \right\}$$
The mapping diagram for this function is shown in the figure.
A function of the form $\left\{ (x,\ y) \right|y = mx + c\}$ is called a linear function because its geometric representation is a straight line.
The equation $y = mx + c$ represents a straight line, where $m\ $is the slope and $\mathbf{c}$ is the $y$-$intercept$.
A function of the form $\{(x,\ y)\ |\ y = ax^{2} + bx + c\}$ is called a quadratic function. It represents a parabolic curve in geometric representation.