Complete Reference Guide for Class 9 Mathematics
Prepared by Muhammad Tayyab (SSE Math)
Govt. Christian High School Daska
Statements · Truth Tables · Logical Reasoning · Deductive & Inductive Proofs
Class 9 Mathematics | Punjab Curriculum and Textbook Board Syllabus 2025
Logic is a systematic method of reasoning that enables us to interpret the meanings of statements, examine their truth, and deduce new information from existing facts.
It plays a key role in problem-solving and decision-making.
Note: The study of logic begins with understanding a statement, which is a sentence that is either true or false, but not both.
Inductive reasoning is when we make a general conclusion from repeated observations or experiences.
Example: A person receives a penicillin injection once or twice and experiences a reaction. He concludes that he is allergic to penicillin.
Deductive reasoning is when we draw a conclusion from already known or accepted facts.
Example: All men are mortal. We are men. So, we are mortal.
A statement is a sentence or mathematical expression that is either true or false, but not both.
Example: The equation \( a = b \) is a statement, it can either be true or false, depending on the values of \( a \) and \( b \).
Note: We can think of a mathematical statement as a unit of information that is either accurate or inaccurate.
The letters \( p \), \( q \) etc., are used to denote statements. A brief list of commonly used logical symbols is given below:
| Symbol | How to be read | Symbolic Expression | How to be read |
|---|---|---|---|
| \( \sim \) | Not | \( \sim p \) | Not \( p \), negation of \( p \) |
| \( \land \) | And | \( p \land q \) | \( p \) and \( q \) |
| \( \vee \) | Or | \( p \vee q \) | \( p \) or \( q \) |
| \( \rightarrow \) | If... then..., implies | \( p \rightarrow q \) | If \( p \) then \( q \), \( p \) implies \( q \) |
| \( \leftrightarrow \) | if and only if, Is equivalent to | \( p \leftrightarrow q \) | \( p \) if and only if \( q \), \( p \) is equivalent to \( q \) |
If \( p \) is any statement, its negation is denoted by \( \sim p \), read as "not \( p \)".
It follows from this definition that:
This relationship is shown in Truth Table.
The conjunction of two statements \( p \) and \( q \) is symbolically written as \( p \land q \) (read as "\( p \) and \( q \)").
A conjunction is considered to be true only if both statements are true.
This relationship is shown in Truth Table.
Solution:
(i) is true (both parts are true).
(ii) is true (both inequalities are true).
(iii) is false (both mathematical statements are false).
The disjunction of statements \( p \) and \( q \) is symbolically written as \( p \vee q \) (read as "\( p \) or \( q \)").
A disjunction is true when at least one of the statements is true. It is false only when both statements are false.
This relationship is shown in Truth Table.
Both statements are true, so the disjunction is true.
Both statements are false, so the disjunction is false.
A compound statement of the form "if \( p \) then \( q \)" (symbolically written as \( p \rightarrow q \)) is called a conditional or an implication. It may also be read as "\( p \) implies \( q \)".
A conditional statement \( p \rightarrow q \) is considered to be:
Statement: If person A lives in Lahore, then he lives in Pakistan.
Let us examine all possibilities:
The statement \( p \rightarrow q \land q \rightarrow p \) is shortly written as \( p \leftrightarrow q \) and is called the biconditional or equivalence. It is read as "\( p \) if and only if \( q \)".
From the truth table, it appears that \( p \leftrightarrow q \) is true only when both statements \( p \) and \( q \) are true or both are false.
Let \( p \) and \( q \) be statements, and let \( p \rightarrow q \) be a given conditional. Then:
The truth values of the Conditional, Converse, Inverse, and Contrapositive are shown in the truth table:
| \( p \) | \( q \) | \( \sim p \) | \( \sim q \) | \( p \rightarrow q \) | \( q \rightarrow p \) | \( \sim p \rightarrow \sim q \) | \( \sim q \rightarrow \sim p \) |
|---|---|---|---|---|---|---|---|
| \( T \) | \( T \) | \( F \) | \( F \) | \( T \) | \( T \) | \( T \) | \( T \) |
| \( T \) | \( F \) | \( F \) | \( T \) | \( F \) | \( T \) | \( T \) | \( F \) |
| \( F \) | \( T \) | \( T \) | \( F \) | \( T \) | \( F \) | \( F \) | \( T \) |
| \( F \) | \( F \) | \( T \) | \( T \) | \( T \) | \( T \) | \( T \) | \( T \) |
| \( p \) | \( q \) | \( p \rightarrow q \) | \( (p \rightarrow q) \land p \) | \( \left[(p \rightarrow q) \land p\right] \rightarrow q \) |
|---|---|---|---|---|
| \( T \) | \( T \) | \( T \) | \( T \) | \( T \) |
| \( T \) | \( F \) | \( F \) | \( F \) | \( T \) |
| \( F \) | \( T \) | \( T \) | \( F \) | \( T \) |
| \( F \) | \( F \) | \( T \) | \( F \) | \( T \) |
Let \( U \) be the universal set. Consider the conditional:
\( \forall x \in U, \quad x \in \phi \rightarrow x \in A \)
The antecedent of this conditional is false because no \( x \in U \) is a member of \( \phi \). Hence, the conditional is true.
In daily life, we often need to prove our claims using solid evidence.
Example 1: If a student named Fayyaz comes home late, his father may doubt he attended school. Just saying "I went to school" is not enough—he needs proof, like:
Example 2: If your mobile phone breaks down and it's under warranty, the company will ask for a warranty card as proof before fixing it.
In mathematics, a proof is a step-by-step logical explanation that shows a statement is true.
It is like evidence that supports a mathematical claim.
Note:
If \( x \) is an odd integer, then it can be expressed in the form:
\( x = 2k + 1, \quad \text{for some } k \in \mathbb{Z} \)
If \( x \) is an even integer, then it can be expressed in the form:
\( x = 2k, \quad \text{for some } k \in \mathbb{Z} \)
(a) Solution:
Let \( x \) be an odd integer. By definition, an odd number can be written as:
\( x = 2k + 1, \quad \text{where } k \in \mathbb{Z} \)
Now taking square both sides:
\( x^{2} = (2k + 1)^{2} \)
\( x^{2} = (2k)^{2} + 2(2k)(1) + (1)^{2} \)
\( x^{2} = 4k² + 4k + 1 \)
\( x^{2} = 2(2k² + 2k) + 1 \)
Let \( m = 2k² + 2k \), which is an integer (since \( k \in \mathbb{Z} \)), so:
\( x^{2} = 2m + 1 \)
This is the standard form of an odd number. Therefore, \( x^{2} \) is an odd integer, by definition.
(b) Solution:
Let \( x \) and \( y \) be odd integers. By definition:
\( x = 2k + 1 \quad \text{where } k \in \mathbb{Z} \)
and \( y = 2n + 1, \quad \text{where } n \in \mathbb{Z} \)
Now adding both:
\( x + y = (2k + 1) + (2n + 1) \)
\( x + y = 2k + 2n + 2 \)
\( x + y = 2(k + n + 1) \)
Let \( m = k + n + 1 \), which is an integer.
So:
\( x + y = 2m \)
This is the standard form of an even number. Therefore, the sum of two odd integers is an even integer, by definition.
(i) Solution:
Let \( x \in (A \cup B)' \)
\( \Rightarrow x \notin (A \cup B) \)
\( \Rightarrow x \notin A \text{ and } x \notin B \)
\( \Rightarrow x \in A' \text{ and } x \in B' \)
\( \Rightarrow x \in A' \cap B' \)
Since \( x \in (A \cup B)' \) was arbitrary, we conclude:
\( (A \cup B)' \subseteq A' \cap B' \quad \ldots (i) \)
Now suppose \( y \in A' \cap B' \)
\( \Rightarrow y \in A' \text{ and } y \in B' \)
\( \Rightarrow y \notin A \text{ and } y \notin B \)
\( \Rightarrow y \notin (A \cup B) \)
\( \Rightarrow y \in (A \cup B)' \)
Thus:
\( A' \cap B' \subseteq (A \cup B)' \quad \ldots (ii) \)
From \( (i) \) and \( (ii) \), we conclude:
\( (A \cup B)' = A' \cap B' \)
(ii) Solution:
Let \( x \in (A \cap B)' \)
\( \Rightarrow x \notin (A \cap B) \)
\( \Rightarrow x \notin A \text{ or } x \notin B \)
\( \Rightarrow x \in A' \text{ or } x \in B' \)
\( \Rightarrow x \in A' \cup B' \)
Since \( x \in (A \cap B)' \) was arbitrary, we conclude:
\( (A \cap B)' \subseteq A' \cup B' \quad \ldots (i) \)
Now suppose \( y \in A' \cup B' \)
\( \Rightarrow y \in A' \text{ or } y \in B' \)
\( \Rightarrow y \notin A \text{ or } y \notin B \)
\( \Rightarrow y \notin (A \cap B) \)
\( \Rightarrow y \in (A \cap B)' \)
Thus:
\( A' \cup B' \subseteq (A \cap B)' \quad \ldots (ii) \)
From \( (i) \) and \( (ii) \), we conclude:
\( (A \cap B)' = A' \cup B' \)
A theorem is a mathematical statement that has been proved to be true using logical steps based on previously accepted facts, definitions, or other theorems.
The sum of interior angles of any quadrilateral is \( 360^\circ \).
Every integer greater than 1 can be uniquely written as a product of prime numbers, ignoring the order of the factors. OR
Every whole number greater than 1 can be written as a product of prime numbers, in only one way (ignoring order).
There are no positive numbers \( a, b, c \) such that
\( a^{n} + b^{n} = c^{n} \text{ for any } n > 2. \)
This statement was made by Pierre Fermat, a French mathematician from the 17th century.
A conjecture is a mathematical statement that is believed to be true, based on observations, but not yet proven.
The Goldbach Conjecture says: Every even number greater than 2 is the sum of two prime numbers. For example,
\( 4 = 2 + 2 \)
\( 6 = 3 + 3 \)
\( 12 = 5 + 7 \)
Although no one has found an even number that violates this, the conjecture has not yet been proved. It remains one of the oldest unsolved problems in mathematics.
An axiom is a basic mathematical fact that is accepted as true without proof. It forms the foundation of mathematics.
Example: Through a point, infinitely many lines can pass. We accept it as true based on intuition and experience.
Both are assumed to be true without proof.
A deductive proof is a method of proving a statement by using logical reasoning from facts that are already known to be true. For example,
Premise 1: All human beings need to breathe to live.
Premise 2: Ahmad is a human.
Conclusion: Therefore, Ahmad needs to breathe to live.
| Conditional | Converse | Inverse | Contrapositive |
|---|---|---|---|
| \( \sim p \rightarrow q \) | \( q \rightarrow \sim p \) | \( p \rightarrow \sim q \) | \( \sim q \rightarrow p \) |
| \( q \rightarrow p \) | \( p \rightarrow q \) | \( \sim q \rightarrow \sim p \) | \( \sim p \rightarrow \sim q \) |
| \( \sim p \rightarrow \sim q \) | \( \sim q \rightarrow \sim p \) | \( p \rightarrow q \) | \( q \rightarrow p \) |
| \( \sim q \rightarrow \sim p \) | \( \sim p \rightarrow \sim q \) | \( q \rightarrow p \) | \( p \rightarrow q \) |
| Mathematical Statement | Mathematical Proof |
|---|---|
| A mathematical statement is a sentence or mathematical expression that is either true or false but not both. | A proof is a logical explanation that verifies a statement using definitions, axioms, known theorems, or logical reasoning. |
| (i) If \( x \) is an odd integer, then \( x^{2} \) is also an odd integer. | (i) If \( x = 2k + 1 \Rightarrow x^{2} = 2m + 1 \) → Odd (proved using algebra). |
| (ii) The sum of two odd numbers is an even number. | (ii) If \( x = 2k + 1, y = 2n + 1 \Rightarrow x + y = 2m \) → Even (proved using identities). |
| Axiom | Theorem |
|---|---|
| An axiom is a mathematical statement accepted without proof. It is based on basic facts or everyday experience and forms the foundation of further reasoning. | A theorem is a mathematical statement that has been proven true using axioms, previously known theorems, and logical steps. |
| (i) Through a given point, infinitely many lines can pass. | (i) The sum of interior angles of a quadrilateral is 360°. |
| (ii) Euclid's Axiom: A straight line can be drawn between any two points. | (ii) Fundamental Theorem of Arithmetic: Every integer greater than 1 can be expressed uniquely as a product of primes. |
Logical reasoning is essential in mathematics because it allows us to prove or disprove statements using facts, definitions, and structured thinking. Without logical reasoning, we cannot be certain whether a mathematical statement is valid.
Consider this deductive reasoning:
This same type of reasoning is used in mathematical deductive proofs. For example, in algebra, we use identities and rules to show that both sides of an equation are equal through step-by-step logic. This ensures the result is always true if the premises are true.
| Statement | Type | Reasoning |
|---|---|---|
| (i) There is exactly one straight line through any two points. | Axiom | This is a basic assumption in Euclidean geometry, accepted without proof. |
| (ii) Every even number greater than 2 can be written as the sum of two primes. | Conjecture | This is the Goldbach Conjecture, believed to be true but not yet proven. |
| (iii) The sum of the angles in a triangle is \( 180^\circ \) | Theorem | This is a proven result in Euclidean geometry, based on axioms and logical reasoning. |
Click on any chapter to view its definitions and formulas