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Introduction

Introduction

Mathematical reasoning is the concept in Mathematics that deals with finding the truth values of any Mathematical statement.  The principle of Mathematical reasoning is generally used to analyse the conceptual logical thinking capacity of an individual in competitive examinations and eligibility tests. Mathematical reasoning questions are extremely interesting and successfully stir up the rational thinking of the human brain. There are various kinds of statements in Mathematical reasoning and the operations that are performed on those statements.


Important Terminologies in Mathematical Reasoning

  • Statement

  • Truth value of a statement 

  • Truth Table

  • Negation operation

  • Compound statements

  • Conditional Statements

  • Contrapositive Statement

  • Tautology

  • Algebra of Statements


Compound Statement and Basic logical Statement

If two or more simple statements are combined by the words ‘and’, ‘or’, ‘if…..then’, ‘if and only if’ then the resultant statement is called a compound statement and words ‘and’, ‘or’ , ‘if…..then’, ‘if and only if’ are called logical connectives.


Examples: (i) Yuvi studies in class XII and resides in Himachal Pradesh.


(ii) Vineet might be watching a movie or playing chess

(iii) India can win the match if and only if Sachin hits a six on each of the remaining 3 balls.


Basic Logical Connective 

Symbol 

Compound Statement

Example

And

$\wedge$

Conjunction

$p\wedge q$

Or

$\vee$

Disjunction

$p\vee  q$

If…..then

$\rightarrow$

Conditional statement

$p\rightarrow q$

If and only if

$\leftrightarrow$

Biconditional statement

$p\leftrightarrow  q$ 


Truth Table of Compound Statements Using Connective ‘AND’ and ‘OR’

Truth Values of statement ‘ p and q’ and ‘p or q’ are tabulated as follows:


$p$

$q$

$p\wedge q$

$p\vee  q$

T

T

T

T

T

F

F

T

F

T

F

T

F

F

F

F


  • Statement ‘p and q’ is true when both p and q are true. Also, this statement is false if at least one of the statements is false.

  • Statement ‘p or q’ is true if at least one of the statements is true. Also, this statement is false if both statements are false.


Conditional Statements

Consider the following compound statements:

If Varanasi is in India then 3 + 6 = 9.

If Varanasi is in India then 3 + 6 = 11

If Varanasi is in China then 3 + 6 = 9.

If Varanasi is in India then 3 + 6 = 11.

If the first statement is p and the second statement is q, then we have the following truth table for $p\rightarrow q$.


$p$

$q$

$p\rightarrow q$

T

T

T

T

F

F

F

T

T

F

F

T


Biconditional Statements:

Consider the following compound statements

If Varanasi is in India if and only if 3 + 6 = 9.

If Varanasi is in India if and only if 3 + 6 = 11

If Varanasi is in China if and only if 3 + 6 = 9.

If Varanasi is in India if and only if 3 + 6 = 11.


If the first statement is p and the second statement is q, then the following truth table for $p\leftrightarrow  q$.


$p$

$q$

$p\leftrightarrow  q$

T

T

T

T

F

F

F

T

F

F

F

T


Logical Equivalence

Two compound statements $S_{1}\left ( p,\,q,\,r,\,.... \right )$ and $S_{2}\left ( p,\,q,\,r,\,.... \right )$ of component statement p, q, r …. are called logically equivalent or simply equivalent or equal if they have identical truth values and we can write 

$\Rightarrow S_{1}\left ( p,\,q,\,r,\,.... \right )\equiv S_{2}\left ( p,\,q,\,r,\,.... \right )$

Note:  

  • $p\to q\equiv \left ( \sim p \right )\vee q$

  • $p\leftrightarrow q\equiv \left ( p\to q \right )\wedge \left ( q\rightarrow p \right )$

  • $p\leftrightarrow q\equiv \left ( \sim p\vee q \right )\wedge \left ( p\,\,\vee \sim q \right )$


Converse, Inverse and Contrapositive of Statement

Given an if-then statement “if p, then q” we can create three related statements.

A conditional statement consists of two parts, a hypothesis in the “ if'' clause and a conclusion in the “then” clause.

For instance, consider the compound statement, 

“If it rains, then they cancel the picnic.”

Here, 

“It rains” is the hypothesis.

“They cancel the picnic” is the conclusion.

  • To form the converse of the conditional statement, just interchange the hypothesis and the conclusion. The converse of the statement (1) is “if they cancel the picnic, it rains.”

  • To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. The inverse of the statement (1) is “if it does not rain, then they do not cancel the picnic.”

  • To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. The contrapositive of the statement (1) is “If they do not cancel the school, then it does not rain.”


Statement 

If p then q $\left ( p\rightarrow q \right )$

Converse

If q then p $\left ( q\rightarrow p \right )$

Inverse

If not p then not q $\left ( \sim p\rightarrow \sim q \right )$

Contrapositive

If not q then not p $\left ( \sim q\rightarrow \sim p \right )$


Tautology and Contradiction (Fallacy)

  • Tautology: A compound statement is called a tautology if it is always true for all possible truth values of its component statements

  • Contradiction: A compound statement is called a contradiction if it is always false for all possible values of its component statement. A contradiction is also called a fallacy.

  • The negation of a tautology is a contradiction and negation of contradiction is a tautology.


Algebra of Statements

  • Idempotent Laws: If p is any statement, then 

(i). $p\vee p\equiv p$            (ii) $p\wedge  p\equiv p$

  • Complement Laws: If p is any statement, then

(i). $p\,\vee \sim p\equiv t$       (ii) $p\,\,\wedge \sim p\equiv f$    

(iii) $\sim\sim p\equiv p$           (iv) $\sim t\equiv f$ 

(v) $\sim f\equiv t$ 

  • Identity Laws: If p is any statement, then 

(i) $p\wedge  t\equiv p$              (ii) $p\wedge  f\equiv f$

(iii) $p\vee t\equiv t$                   (iv) $p\vee f\equiv p$

  • Commutative Laws: If p and q be any two statements then

(i) $p\wedge q\equiv q\wedge p$     (ii) $p\vee q\equiv q\vee  p$

  • De Morgan’s Laws: If p and q be any two statements, then 

(i) $\sim\left ( p\vee q \right )\equiv \sim p\,\,\wedge \sim q$ 

(ii) $\sim\left ( p\,\ \wedge  q \right )\equiv \sim p\,\,\vee  \sim q$ 

  • Associative Laws: If p, q and r are any three statements, then 

(i) $\left ( p\wedge q \right ) \wedge r\equiv p\wedge \left ( q\wedge r \right )$

(ii) $\left ( p\vee q \right )\vee r\equiv p\vee \left ( q\vee r \right )$

  • Distributive Laws: If p, q and r are any three statements, then

(i) $p\wedge \left ( q\vee r \right )\equiv \left ( p\wedge q \right )\vee \left ( p\wedge r \right )$

(ii) $p\vee \left ( q\wedge r \right )\equiv \left ( p\vee q \right )\wedge \left ( p\vee r \right )$


Solved Examples:

Example 1: Show that the compound statements $\left ( p\vee q \right )\wedge \sim p$ and $\sim p \wedge q$ are logically equivalent.

Solution: Truth values of $\left ( p\vee q \right )\wedge \sim p$ and $\sim p \wedge q$ are as follows:


$p$

$q$

$\sim p$

$p\vee q$

$\left (  p\vee q\right )\wedge \sim p$

$\sim p\,\,\wedge q$

T

T

F

T

F

F

T

F

F

T

F

F

F

T

T

T

T

T

F

F

T

F

F

F


Clearly, $\left ( p\vee q \right )\wedge \sim p$ and $\sim p \wedge q$ have identical truth values.

$\therefore \left ( p \vee q \right )\wedge \sim p \equiv \,\,\sim p\wedge q$ 


Example 2: Show that $p\to \left ( p\vee q \right )$ is a tautology

Solution: Truth values of $p\to \left ( p\vee q \right )$

$p$

$q$

$p\vee q$

$p\to \left ( p\vee q \right )$

T

T

T

T

T

F

T

T

F

T

T

T

F

F

F

T


Thus, for all possible truth values of p and q, the compound statement $p\to \left ( p\vee q \right )$ is true.


Example 3: Prove that $q\,\,\wedge \sim p\equiv \sim \left ( q\to p \right )$

Solution: $\sim\left ( q\to p \right ) \equiv \,\sim \left ( \sim q\vee p \right )$

$\Rightarrow \sim\left ( q\to p \right ) \equiv \,\sim \left ( \sim q \right )\wedge \sim p$

$\Rightarrow \sim\left ( q\to p \right ) \equiv \, q\,\,\wedge \sim p$


Solved problems of Previous Year Question

1. Which of the following is tautology?

  1. $A\wedge \left ( A\to B \right )\to B$

  2. $B\to \left ( A\wedge A\to B \right )$

  3. $A\wedge \left ( A\vee B \right )$

  4. $\left (A\vee B  \right )\wedge A$

Ans: (a)

$\Rightarrow A\wedge \left ( A\to B \right ) \to B$

$= A\wedge \left ( \sim A\vee B \right )\to B$

$=\left [ \left ( A\wedge \sim A \right )\vee \left ( A\wedge B \right ) \right ]\to B$

$=\left ( A\wedge B \right )\to B$

$=\,\, \sim A\,\,\vee \sim B\vee B$

$= t$


2. The Boolean expression $\sim\left ( p\vee q \right )\vee \left ( \sim p\vee q \right )$ is equivalent to 

  1. $\sim q$

  2. $\sim p$

  3. $p$

  4. $q$

Ans: (b)

$\Rightarrow \,\,\sim \left ( p\vee q \right )\vee \left ( \sim p\wedge q \right )$

$= \left ( \sim p\,\,\wedge \sim q \right )\vee \left ( \sim p\wedge q \right )$

$=\sim p\wedge \left ( \sim q\wedge q \right )$

$=\,\,\sim p$


3. The statement $\sim \left ( p\leftrightarrow \,\sim q \right )$ is 

  1. equivalent to $p\leftrightarrow q$

  2. equivalent to $\sim p\leftrightarrow q$

  3. a tautology 

  4. a fallacy

Ans: (a)


$p$

$q$

$\sim q$

$p\leftrightarrow \,\,\sim q$

$\sim\left ( p\leftrightarrow \,\,\sim q \right )$

$p\leftrightarrow q$

T

T

F

F

T

T

T

F

T

T

F

F

F

T

F

T

F

F

F

F

T

F

T

T

 

Practice Question:

1. $\sim\left ( p\vee \left ( \sim p\vee q \right ) \right )$ is equal to

  1. $\sim p\wedge \left ( p\,\,\wedge \sim q \right )$

  2. $\left ( p\,\,\wedge \sim q \right )\vee \sim p$

  3. $\left ( p\,\,\vee \sim q \right )\vee \sim p$

  4. None of these


2. $\left ( p\,\,\wedge \sim q \right )\wedge \left ( \sim p\wedge q \right )$ is 

  1. A tautology

  2. A contradiction

  3. Neither a tautology nor a contradiction

  4. None of these


3. If $p\to \left ( q\vee r \right )$ is false, then the truth values of p, q and r are, respectively

  1. T, T, F

  2. F, F, F

  3. F, T, T

  4. T, F, F


Answer: 

  1. (a)

  2. (b)

  3. (d) 


Conclusion:

In this chapter, we have elaborated on concepts and solutions to questions on the topic of Mathematical reasoning.  Everything you're looking for is available in a single location. Students can carefully read through the Concepts, Definitions, and questions in the PDFs, which are also free to download and understand the concepts used to solve these questions. This will be extremely beneficial to the students in their exams.

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Frequently Asked Questions
FAQ

1. What is a fallacy in mathematical reasoning?

Fallacies are errors in assumptions that are induced by logical inaccuracy.

2. What is a boolean expression?

Boolean Algebra is the branch of mathematics that works with only binary values. In contrast to numerical operations like addition and subtraction, a Boolean equation deals with conjunction, disjunction, and negation. Boolean algebra uses binary codes to express the value and carries out logical computations through operations like AND and OR. The simplicity of Boolean expression makes it the best choice for being used in electronics and digital advancements. It would not be wrong to say that it has successfully revolutionised the world of computers.

3. What are Mathematical Statements?

A mathematical statement is the basis of all mathematics quantitative reasoning. Further, reasoning can be either inductive (also called mathematical induction) or deductive. An assertive statement that can either be true or false is said to be an acceptable statement in mathematics. It cannot be both. Ambiguous statements are invalid in maths and reasoning.

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