Unit 8 Class 9 Math Solutions – Logic
Unit 8 Class 9 Math Solutions contain the complete PDF solution of Exercise 8 from the chapter Logic. This unit has only one exercise, so students can view and download the complete Exercise 8 solutions from this page.
The chapter introduces mathematical reasoning, statements, truth values, logical connectives, conditionals, converse, inverse, contrapositive, truth tables, axioms, conjectures, theorems, and direct mathematical proofs.
The PDF includes full questions, carefully explained working, truth tables, algebraic proofs, set proofs, geometric reasoning, and boxed final answers for the Punjab Board Class 9 Mathematics book.
Students can use these Unit 8 solutions for homework, revision, class tests, annual examinations, and board exam preparation.
Solutions of Exercise 8 Unit 8 Class 9 Math Notes
Exercise 8 of Unit 8 Class 9 Math Notes covers Logic and Mathematical Reasoning. It explains how mathematical conclusions are formed, how compound statements are analyzed, and how a claim is established through a valid proof.
Students can view the complete solution PDF on this page and use the download or save option in the PDF viewer to keep a copy on their device.
Inductive Reasoning
Inductive reasoning begins with examples, observations, experiments, or repeated patterns.
After examining several particular cases, a likely general conclusion is formed.
An inductive conclusion may be strongly supported by evidence, but it is not automatically a proof for every possible case.
For example, observing that a pattern works for many values may lead to a conjecture.
Deductive Reasoning
Deductive reasoning begins with accepted facts, definitions, axioms, or previously proved results.
Each step follows logically from the previous step, so the conclusion must be true when the reasoning is valid.
Mathematical proofs mainly use deductive reasoning.
Mathematical Statements
A mathematical statement is a sentence that has a definite truth value.
It must be either true or false.
A sentence that is a question, command, or open expression with an unspecified variable is not a complete mathematical statement until its truth value can be determined.
Truth Values
The truth value of a statement is written as true or false.
In truth tables, these values are usually represented by:
\(T\)
and:
\(F\)
Negation
The negation of a statement denies the original statement.
If the statement is represented by \(p\), its negation is:
\(\sim p\)
The negation is true when the original statement is false, and false when the original statement is true.
Conjunction
The conjunction of two statements is written as:
\(p\land q\)
It means “\(p\) and \(q\).”
A conjunction is true only when both statements are true.
Disjunction
The disjunction of two statements is written as:
\(p\lor q\)
It means “\(p\) or \(q\).”
A disjunction is false only when both statements are false.
Conditional Statement
A conditional statement is written as:
\(p\rightarrow q\)
It means “if \(p\), then \(q\).”
The statement \(p\) is called the antecedent, while \(q\) is called the consequent.
A conditional is false only when the antecedent is true and the consequent is false.
Biconditional Statement
A biconditional statement is written as:
\(p\leftrightarrow q\)
It means “\(p\) if and only if \(q\).”
A biconditional is true when both statements have the same truth value.
Converse of a Conditional
The converse is formed by interchanging the antecedent and consequent.
For:
\(p\rightarrow q\)
the converse is:
\(q\rightarrow p\)
Inverse of a Conditional
The inverse is formed by negating both the antecedent and consequent without changing their order.
For:
\(p\rightarrow q\)
the inverse is:
\(\sim p\rightarrow\sim q\)
Contrapositive of a Conditional
The contrapositive is formed by interchanging the two statements and negating both.
For:
\(p\rightarrow q\)
the contrapositive is:
\(\sim q\rightarrow\sim p\)
A conditional statement and its contrapositive are logically equivalent.
Truth Tables
A truth table lists every possible combination of truth values for the simple statements.
For two statements \(p\) and \(q\), the four possible rows are:
\((T,T),\ (T,F),\ (F,T),\ (F,F)\)
Intermediate columns should be completed before the final compound statement is evaluated.
Truth Table of a Negation
The truth value of a negation is always opposite to the original statement.
\(p=T\Rightarrow\sim p=F\)
\(p=F\Rightarrow\sim p=T\)
Truth Table of a Conjunction
The final column of \(p\land q\) contains true only in the row where both \(p\) and \(q\) are true.
Truth Table of a Disjunction
The final column of \(p\lor q\) contains false only in the row where both \(p\) and \(q\) are false.
Truth Table of a Conditional
The conditional \(p\rightarrow q\) is false only in the case:
\(p=T,\qquad q=F\)
Truth Table of a Biconditional
The biconditional is true when the two parts have matching truth values.
\(T\leftrightarrow T=T\)
\(F\leftrightarrow F=T\)
Mathematical Statement and Proof
A mathematical statement presents a claim that is true or false.
A mathematical proof is a sequence of justified logical steps that establishes why the statement is true.
The statement is the result being claimed, while the proof is its logical justification.
Axiom
An axiom is a basic mathematical statement accepted as true without proof.
For example, one straight line can be drawn through two distinct points.
Axioms provide the starting foundation for mathematical reasoning.
Postulate
A postulate is also accepted without proof and is commonly used in geometry.
In many school-level contexts, the terms axiom and postulate are used in closely related ways.
Theorem
A theorem is a mathematical statement established through proof.
Examples include the Pythagorean theorem and the result that the interior angles of a triangle add to 180 degrees.
Conjecture
A conjecture is a statement supported by examples or evidence but not proved for every case.
A conjecture may later become a theorem if a valid general proof is discovered.
Importance of Logical Reasoning
Logical reasoning ensures that every conclusion follows from known facts.
It prevents unsupported assumptions, helps identify errors, and makes a result valid for every case covered by the proof.
Without logical reasoning, a calculation may look correct without actually establishing the required result.
Direct Proof
A direct proof starts from the given information and proceeds step by step toward the required conclusion.
Definitions, algebraic rules, axioms, and previously proved results are used to justify each step.
Proving Algebraic Identities
To prove an identity, begin with one side, usually the more complicated side.
Apply valid algebraic operations until the other side is obtained.
For example:
\((x+1)^2-(x-1)^2=4x\)
Expanding both squares gives:
\(x^2+2x+1-x^2+2x-1=4x\)
Therefore, the left-hand side equals the right-hand side.
Difference of Two Squares in a Proof
Some identities can be proved more quickly by using:
\(A^2-B^2=(A-B)(A+B)\)
For example:
\((x+5)^2-(x-5)^2\)
becomes:
\([(x+5)-(x-5)][(x+5)+(x-5)]\)
which simplifies to:
\(10\cdot2x=20x\)
Justifying Algebraic Simplification
Every cancellation must be performed on factors, not on separate terms.
Before cancelling, factorize the numerator and denominator completely.
A cancelled factor must be non-zero.
Restrictions on Rational Expressions
Values that make the original denominator zero must remain excluded, even if the corresponding factor is later cancelled.
For example, if the original denominator contains:
\((x-3)(x+3)\)
then:
\(x\neq3,\qquad x\neq-3\)
must remain part of the final result.
Even Integers
An even integer can be written as:
\(x=2n\)
where \(n\) is an integer.
To prove that an expression is even, rewrite it in the form:
\(2k\)
for some integer \(k\).
Odd Integers
An odd integer can be written as:
\(x=2n+1\)
where \(n\) is an integer.
To prove that an expression is odd, rewrite it in the form:
\(2k+1\)
for some integer \(k\).
Proving an Expression Is Odd
Exercise 8 includes direct proofs involving odd integers.
For example, if:
\(x=2n+1\)
then:
\(9x+4=9(2n+1)+4\)
\(=18n+13\)
\(=2(9n+6)+1\)
The final form is odd.
Proving an Expression Is Even
For an odd integer \(x=2n+1\):
\(7x+5=7(2n+1)+5\)
\(=14n+12\)
\(=2(7n+6)\)
The expression is twice an integer and is therefore even.
Polynomial Parity Proofs
The same even and odd forms are used for polynomial expressions.
Substitute \(x=2n\) when \(x\) is even and \(x=2n+1\) when \(x\) is odd.
Simplify until the result is written as \(2k\) or \(2k+1\).
Proving De Morgan’s Law for Sets
Exercise 8 proves the set identity:
\((A\cap B)’=A’\cup B’\)
Let \(x\) be an arbitrary element.
Show that membership in the left-hand side produces exactly the same condition as membership in the right-hand side.
The logical steps are:
\(x\in(A\cap B)’\)
\(\Longleftrightarrow x\notin A\cap B\)
\(\Longleftrightarrow x\notin A\text{ or }x\notin B\)
\(\Longleftrightarrow x\in A’\cup B’\)
Proving Set Equality by Two Inclusions
Another method is to prove:
\((A\cap B)’\subseteq A’\cup B’\)
and:
\(A’\cup B’\subseteq(A\cap B)’\)
When each set is contained in the other, the two sets are equal.
Positive Real Numbers and Their Squares
Exercise 8 includes the result that for positive real numbers:
\(x^2<y^2\Rightarrow x<y\)
Factor the difference:
\(y^2-x^2=(y-x)(y+x)\)
Since \(x\) and \(y\) are positive, \(y+x\) is positive.
Therefore, the other factor must also be positive:
\(y-x>0\)
Hence:
\(x<y\)
Proof of the Angle Sum of a Triangle
A line is drawn through one vertex parallel to the opposite side.
Alternate interior angles are used to transfer the other two triangle angles to the new straight line.
Angles on a straight line add to:
\(180^\circ\)
Therefore:
\(\angle A+\angle B+\angle C=180^\circ\)
Properties of Fractions
Exercise 8 also proves basic properties of fractions, with all denominators non-zero.
The equality of two fractions is connected with cross multiplication:
\(\frac{a}{b}=\frac{c}{d}\Longleftrightarrow ad=bc\)
The product rule is:
\(\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}\)
Fractions with the same denominator are added by adding their numerators:
\(\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}\)
Non-Zero Denominators
A denominator can never be zero.
Restrictions such as:
\(b\neq0,\qquad d\neq0\)
must be stated whenever the proof involves division by those quantities.
Important Rules of Unit 8 Class 9 Math
Negation
\(\sim p\)
Conjunction
\(p\land q\)
Disjunction
\(p\lor q\)
Conditional
\(p\rightarrow q\)
Biconditional
\(p\leftrightarrow q\)
Converse
\(p\rightarrow q\quad\Rightarrow\quad q\rightarrow p\)
Inverse
\(p\rightarrow q\quad\Rightarrow\quad\sim p\rightarrow\sim q\)
Contrapositive
\(p\rightarrow q\quad\Rightarrow\quad\sim q\rightarrow\sim p\)
Even Integer Form
\(x=2n\)
Odd Integer Form
\(x=2n+1\)
Difference of Two Squares
\(A^2-B^2=(A-B)(A+B)\)
De Morgan’s Law for Sets
\((A\cap B)’=A’\cup B’\)
Equality of Fractions
\(\frac{a}{b}=\frac{c}{d}\Longleftrightarrow ad=bc\)
Common Mistakes in Unit 8 Class 9 Math Notes
Students sometimes confuse inductive reasoning with a complete proof.
The converse, inverse, and contrapositive are often written without interchanging or negating the correct statements.
A conditional is not false whenever the consequent is false; it is false only when the antecedent is true and the consequent is false.
Truth-table rows should include every possible combination of truth values.
Students may begin a proof by assuming the exact result that they are required to prove.
In algebraic proofs, terms should not be cancelled unless the expression has first been factorized.
Restrictions from the original denominator must not be removed after cancellation.
To prove that an expression is even, it must be written as twice an integer.
To prove that an expression is odd, it must be written as twice an integer plus one.
In a set proof, an arbitrary element should be followed through both sides of the identity.
A conjecture is supported by evidence but is not the same as a theorem.
Every step in a geometric proof should be justified by a construction, axiom, or known angle property.
Exam Preparation Tips for Unit 8 Class 9 Math Notes
Learn the meanings and symbols of negation, conjunction, disjunction, conditional, and biconditional.
Memorize the converse, inverse, and contrapositive of a conditional.
Practise completing truth tables one column at a time.
Learn the difference between a statement, proof, axiom, conjecture, and theorem.
Practise short direct proofs involving even and odd integers.
Keep denominator restrictions in every rational-expression proof.
Use an arbitrary element when proving a set identity.
Write a reason for every major step of an algebraic or geometric proof.
Attempt the complete Exercise 8 independently before checking the PDF solution.
Use the download or save option in the PDF viewer to keep the solutions for offline revision.
Why Unit 8 Class 9 Math Solutions Are Important
Unit 8 Class 9 Math Solutions are important because logic provides the foundation of mathematical proof.
This chapter teaches students how to distinguish evidence from proof and how to reach conclusions through valid reasoning.
Truth tables improve the understanding of compound statements, while direct proofs strengthen algebraic, set, and geometric reasoning.
These skills are useful in higher mathematics, computer science, programming, statistics, philosophy, and scientific problem-solving.
FAQs About Unit 8 Class 9 Math Solutions
What is the topic of Unit 8 Class 9 Math?
The topic of Unit 8 is Logic and Mathematical Reasoning.
How many exercises are included in Unit 8?
Unit 8 contains only one exercise, which is Exercise 8.
What is covered in Exercise 8?
Exercise 8 covers inductive and deductive reasoning, logical connectives, conditionals, truth tables, axioms, conjectures, theorems, and mathematical proofs.
What is a mathematical statement?
A mathematical statement is a sentence with a definite truth value: it is either true or false.
What is the difference between an axiom and a theorem?
An axiom is accepted without proof, while a theorem is established through proof.
What is a conjecture?
A conjecture is a statement supported by examples or evidence but not proved for every case.
When is a conditional statement false?
The conditional \(p\rightarrow q\) is false only when \(p\) is true and \(q\) is false.
What is the contrapositive of a conditional?
The contrapositive of \(p\rightarrow q\) is \(\sim q\rightarrow\sim p\).
How is an even integer represented?
An even integer is represented by \(2n\), where \(n\) is an integer.
How is an odd integer represented?
An odd integer is represented by \(2n+1\), where \(n\) is an integer.
Can students download the Unit 8 solutions PDF?
Yes. Students can view the PDF on this page and use the save or download option provided by the PDF viewer.
Are these Unit 8 Class 9 Math Solutions useful for exam preparation?
Yes. The solutions are useful for homework, revision, class tests, annual examinations, and board exam preparation.
Disclaimer
These Unit 8 Class 9 Math Solutions are provided for educational help. Students should use them to understand the reasoning and proof methods, check their work, and study alongside the official textbook and their teacher’s instructions.
Final Words
Unit 8 Class 9 Math Solutions help students understand Logic and Mathematical Reasoning in a clear and organized way.
Because this unit contains only Exercise 8, the complete chapter solution can be viewed and downloaded from one PDF on this page.
Study the logical symbols carefully, practise truth tables, and focus on giving a valid reason for every step in a proof.