Mathematical Reasoning is the process of applying logical thinking and mathematical concepts to analyze statements, identify patterns, solve problems, and draw correct conclusions. It helps in determining whether a statement is true or false using facts, rules, and logical arguments.
Example: “The sum of two even numbers is always even” is a mathematically acceptable statement because it has a definite truth value and is always true.
Types of Reasoning in Mathematics
Mathematical reasoning mainly involves two types of reasoning:
These reasoning methods help identify patterns, draw conclusions, and solve mathematical problems.
Inductive Reasoning
Inductive reasoning is the process of observing specific examples or patterns and forming a general conclusion from them. The conclusion is based on observations and may not always be true in every case.
Example:
2 + 4 = 6
4 + 6 = 10
6 + 8 = 14From these examples, we conclude that the sum of two even numbers is always even.
Deductive Reasoning
Deductive reasoning is the process of using known facts, rules, or definitions to reach a logically certain conclusion. If the given statements are true, the conclusion must also be true.
Example:
All squares have four sides.
ABCD is a square.Therefore, ABCD has four sides.
Statements in Mathematical Logic
A statement is a sentence that has a definite truth value, meaning it is either true or false, but not both at the same time. Sentences that are unclear or do not have a definite truth value are not considered statements.
Examples
- “Republic Day is on 26 January.”
This is a true statement.- “The weight of an ant is greater than the weight of an elephant.”
This is a false statement.
Since the truth values of these sentences can be clearly determined, they are considered mathematical statements.
Types of Statements in Mathematical Logic
- Simple statement: Simple statements are those statements whose truth value does not explicitly depend on another statement. They are direct and do not include any modifier.
Example: '364 is an even number'
- Compound statement: When two or more simple statements are combined using the 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' are also called logical connectives.
Example: 'I am studying psychology and history'.
Logical Connectives
- 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,' it is known as a disjunction.
a v b
Here, a and b are two simple statements.
- Conditional statement: When a statement is created by connecting two simple statements using 'if...then,' it is known as a conditional statement.
a → b
Here, a and b are two simple statements.
- Biconditional statement: When a statement is created by connecting two simple statements using 'if and only if,' it is known as a biconditional statement.
a ↔ b
Here, a and b are two simple statements.
- Negation: When a statement is created by using words like 'no' or 'not,' it 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.
Value of a statement
The truth value of a statement tells whether the statement is true or false. A true statement is represented by T, while a false statement is represented by F.
Examples
- “364 is an even number.”
This statement is true, so its truth value is T.- “71 is divisible by 2.”
This statement is false, so its truth value is F.
Converse, Inverse and Contrapositive of a Conditional Statement
For a conditional statement (p → q), there are three related statements.
Converse: The converse of a conditional statement is obtained by interchanging the hypothesis and conclusion.
Formula: (q → p)
Inverse: The inverse of a conditional statement is obtained by negating both the hypothesis and conclusion.
Formula: (~p → ~q)
Contrapositive: The contrapositive of a conditional statement is obtained by interchanging and negating both the hypothesis and conclusion.
Formula: (~q → ~p)
Truth table:
As we know, 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, it 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
New Statements from Old Statement
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 the 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. If 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.