# Statements – Mathematical Reasoning

The study of logic through mathematical symbols is called mathematical reasoning. Mathematical logic is also known as Boolean logic. Or in other words, in mathematical reasoning, we determine the truth value of the statement. 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 type of reasoning the validity of the statement is checked using some set of rules and then generalized the given statement. Or 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 true if the assumptions entering the deduction are true. In mathematics, deductive reasoning is more important than inductive reasoning.

**Statements in Mathematical logic**

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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 as 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. In mathematics reasoning, there are two major types of statements are present:

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

(v)7 + 5 = 21False, this will compute to 12 but the answer is given 21 so the answer is false.

**Value of a statement:**

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 2^{n }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:

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

pqp ^ qT 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~pT 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:

pq(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)]

**Examples: **

**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⇒∼qp⇒q⇔∼q⇒∼pF T F T F F T F F F T F Therefore, it is a fallacy.