Statements – Mathematical Reasoning

Statements – Mathematical Reasoning: The study of logic through mathematical symbols is called mathematical reasoning. Mathematical logic is also known as Boolean logic. In other words, in mathematical reasoning, we determine the statement’s truth value.

Mathematical reasoning is of seven types i.e., intuition, counterfactual thinking, critical thinking, backward induction, inductive reasoning, deductive reasoning, and abductive induction. Out of these 7 types, the following two types are the major types:

• Inductive reasoning: In this reasoning, the validity of the statement is checked using some set of rules, and then the given statement is generalized. In other words, inductive reasoning is non-rigorous reasoning, in which the statements are generalized.
• Deductive reasoning: It is rigorous reasoning, in which the statements are assumed to be true if the assumptions entering the deduction are true. In mathematics, deductive reasoning is more important than inductive reasoning.

A sentence is a statement if it is either correct or incorrect or true or false, but it can never be both because a statement that is both true or false cannot be considered a statement and if a sentence is neither true nor false then also it cannot be considered as a statement. Statements are the basic unit of reasoning. For example, we have three statements:

Sentence 1: Republic day is on 26 January

Sentence 2: The weight of ant is greater than the weight of the elephant. 

So, by reading these statements we immediately conclude that sentence 1 is true and sentence 2 is false. Hence, these sentences are accepted as statements because they are either true or false, they are not ambiguous.

Inductive Reasoning:

• Inductive reasoning involves making generalizations based on observed patterns or examples. It starts with specific observations and derives general principles or hypotheses that apply to all similar cases.
• While inductive reasoning doesn’t guarantee absolute certainty, it can provide strong evidence for hypotheses.

Abductive Reasoning:

• Abductive reasoning involves making educated guesses or hypotheses to explain observed phenomena.
• It’s often used in problem-solving when multiple possible explanations exist, and the goal is to identify the most plausible or likely explanation based on available evidence.

Analytical Reasoning:

• Analytical reasoning involves breaking down complex problems into smaller, more manageable parts and analyzing each part individually.
• It emphasizes logical thinking, systematic problem-solving strategies, and the use of mathematical tools and techniques to derive solutions.

Critical Reasoning:

• Critical reasoning involves evaluating arguments, claims, or solutions carefully and objectively, considering their logical validity, coherence, and relevance.
• It emphasizes the ability to identify assumptions, recognize fallacies, and assess the strength of evidence and reasoning presented.

Constructive Reasoning:

• Constructive reasoning involves building new mathematical objects, structures, or proofs by combining existing ones through logical operations or methods.
• It’s often used in constructive mathematics and proof theory to demonstrate the existence of mathematical objects by explicitly constructing them.

Geometric Reasoning:

• Geometric reasoning involves using geometric principles, properties, and relationships to solve problems and prove geometric theorems.
• It often involves visualizing geometric figures, applying geometric formulas, and reasoning about spatial configurations.

Probabilistic Reasoning:

• Probabilistic reasoning involves assessing the likelihood or probability of different outcomes based on available evidence, assumptions, or prior knowledge.
• It’s used in probability theory, statistics, and decision-making to quantify uncertainty and make informed judgments or predictions.

• Simple statement: Simple statements are those statements whose truth value does not explicitly depend on another statement. They are direct and does not include any modifier.

Example:

‘364 is an even number’

• Compound statement: When two or more simple statements are combined using words ‘and’, ‘or’, ‘if…then’, and ‘if and only if’ then the resultant statement is known as a compound statement. ‘and’, ‘or’, ‘if…then’, and ‘if and only if’ these are also called as logical connectives.

Example:

 ‘ I am studying psychology and history’.

Elementary operation of logic:

• Conjunction: When a compound statement is created using ‘and’ is known as a conjunction.

a ^ b

Here, a and b are two simple statements.

• Disjunction: When a compound statement is created using ‘or’ is known as disjunction.

a v b

Here, a and b are two simple statements.

• Conditional statement: When a statement is created by connection two simple statements using ‘if….then’ is known as conditional statement.

a → b

Here, a and b are two simple statements.

• Biconditional statement: When a statement is created by connection two simple statements using ‘if and only if’ is known as biconditional statement.

a  ↔ b

Here, a and b are two simple statements.

• Negation: When a statement is created by using words like ‘no’, ‘not’ is known as negation.

~a

Examples:

Are the following sentences statements? answer in true or false

(i) ”7 + 5 = 19” 

This statement will be considered false because the addition is not correct

(ii) “today’s weather is very nice” 

This statement is neither true nor false because if the weather seems nice to one person it does not mean that every person will share the same opinion.

(iii)” 2 + 5 – 3 + 2 = 6 ” 

This statement will be considered true because the equation is correct.

(iv) harsh is very nice

False, because this is an opinion of a person and opinions can vary.

(v) 7 + 5 = 21

False, this will compute to 12 but the answer is given 21 so the answer is false.

A statement if is either correct or incorrect or true or false. The true or false state of a statement is known as a truth value. If the statement is false it is determined as ‘F’ and if the statement is true it is determined as ‘T’.

Example:

(i) ‘364 is an even number’ is T because this statement is true.

(ii) ’71is divisible by 2′ is F because this statement is false.

Truth table:

As we know that a statement can be true or false and these values are known as truth values. So, a truth table is a summary of the truth values of the resultant statement for all possible combinations of truth values of component statements.

In the case of n number of statements, there are 2ⁿ distinct possible arrangements of truth values in the table of the statements. In the truth table, when the compound statement is true for every condition then it is known as a tautology, and when the compound statement is false for every condition is known as a fallacy.

Example:

The truth table for one statement ‘p’ will be written as:

p
T
F

The truth table of two statements ‘p’ and ‘q’ will be taken as:

p	q	p ^ q
T	T	T
T	F	F
F	T	F
F	F	F

In mathematical reasoning, a new statement is created from the old statement by the negation of the old statement.

Negation of Statements:

If ‘p’ is a statement then the denial of the statement is known as negation. The negation of a statement is denoted by putting a ‘~’ in front of the statement the negation of ‘p’ is ‘~p’. This symbol is defined as that when a symbol is negated the word ‘not’ is inserted in the statement, or we can start the statement by saying ‘It is false that….’

Example:

The truth table will be as:

p	~p
T	F
F	T

Negation of compound statements:

When two or more simple statements are combined using words ‘and’, ‘or’, ‘if…then’, and ‘if and only if’ then the resultant statement is known as a compound statement. So to negate a compound statement we use ‘not’ words. For example, to negate a statement of the form “If P, then Q” we should replace it with the statement “P and Not Q”.

DeMorgan’s Laws: negating compound statements

∼(p ^ q) ↔ (∼p ∨ ∼q)

∼(p ∨ q) ↔ (∼p ^∼q)

(i) Negating conjunction and disjunction

∼(p ^ q) ↔ (∼p ∨ ∼q)

∼(p ∨ q) ↔ (∼p ^∼q)

Examples: 

(i) p ^ q = I will buy snacks and sweets

∼(p ^ q) = I will not buy snacks and sweets

(ii) p v q = I will but headphones or earphones

∼(p v q) = it is not the case that I will be buying headphones or earphones.

(iii) Negate (p ^ q) using truth tables:

p	q	(p ^ q)	∼(p ^ q)
T	T	T	F
T	F	F	T
F	T	F	T
F	F	F	T

(ii) Negating a conditional statement

∼(p → q) = (p ∧ ∼q)

Example:

p → q = if it rains today then i will go to school 

∼(p → q)  –  it is not the case that if rains today, then i will go to schools

(iii) Negating a biconditional statement

∼(p ↔ q) ↔ [(p ∧ ∼q) ∨ (q ∧ ∼p)]

Example

Question 1: Consider the following statements negate the following statements

P: harsh lives is Delhi

Q: harsh is rich

R: harsh is emotionally strong

Solution: 

∼(Q ↔ (P ^ ∼R)

harsh lives in Delhi and is not emotionally strong if and only if harsh is rich.

Question 2. p and q are two statements, then what will be (p ⇒ q) ⇔ (~q ⇒ ~p) show the result in the form of a truth table.

Solution:

p⇒q	∼p⇒∼q	p⇒q⇔∼q⇒∼p
F	T	F
T	F	F
T	F	F
F	T	F

Therefore, it is a fallacy.

Articles related to Mathematical Reasoning:

• Conditional Statements & Implications – Mathematical Reasoning
• Principle of Mathematical Induction
• Introduction to Mathematical Logic
• Number Series Reasoning Questions and Answers
• Introduction to Mathematical Logic

Mathematical reasoning not only enhances our understanding of mathematical concepts but also equips us with essential skills for navigating the challenges of the modern world, from interpreting data and making predictions to analyzing arguments and evaluating evidence. As such, fostering mathematical reasoning skills is integral to fostering intellectual curiosity, creativity, and lifelong learning.

What is mathematical reasoning?

Mathematical reasoning refers to the process of making logical deductions, drawing conclusions, and solving problems using mathematical principles, rules, and methods.

Why is mathematical reasoning important?

Mathematical reasoning is essential for understanding and applying mathematical concepts, solving problems across various domains, and developing critical thinking skills that are valuable in everyday life and academic pursuits.

What are the types of mathematical reasoning?

Types of mathematical reasoning include deductive reasoning, inductive reasoning, abductive reasoning, analytical reasoning, critical reasoning, constructive reasoning, geometric reasoning, and probabilistic reasoning, among others.

How does deductive reasoning differ from inductive reasoning?

Deductive reasoning starts with general principles or premises and derives specific conclusions, following a logical sequence of steps. Inductive reasoning starts with specific observations and derives general principles or hypotheses based on patterns or examples.

What is critical reasoning in mathematics?

Critical reasoning in mathematics involves evaluating arguments, claims, or solutions carefully and objectively, considering their logical validity, coherence, and relevance. It emphasizes identifying assumptions, recognizing fallacies, and assessing the strength of evidence and reasoning.

How can I improve my mathematical reasoning skills?

You can improve your mathematical reasoning skills by practicing problem-solving regularly, exploring a variety of mathematical problems and puzzles, studying mathematical proofs and logical reasoning techniques, and seeking feedback and guidance from teachers or peers.

What role does mathematical reasoning play in everyday life?

Mathematical reasoning helps in making informed decisions, solving practical problems, understanding patterns and trends in data, interpreting quantitative information, and evaluating arguments or claims that involve mathematical concepts.

How does mathematical reasoning contribute to academic success?

Mathematical reasoning is a foundational skill in mathematics education, contributing to success in mathematics courses, standardized tests, and academic disciplines that require quantitative reasoning and problem-solving abilities.

Are there any resources available to help develop mathematical reasoning skills?

Yes, there are many resources available, including textbooks, online courses, problem-solving books, math competitions, educational websites, and mathematical puzzles and games, which can help enhance mathematical reasoning skills.

Can mathematical reasoning be applied outside of mathematics?

Yes, mathematical reasoning skills, such as critical thinking, logical analysis, and problem-solving, are valuable in various fields beyond mathematics, including science, engineering, economics, computer science, and decision-making in everyday life.