Mathematical Reasoning

Mathematical Reasoning, often referred to as deductive reasoning, is a fundamental aspect of Mathematics. It's the process of logically analyzing mathematical statements and drawing conclusions based on established rules and principles. In essence, mathematical reasoning involves constructing logical arguments to prove mathematical statements or theorems.

1.0What is Mathematical Reasoning

Mathematical Reasoning, a crucial component of mathematics, involves the evaluation of the truth values of given statements. These reasoning tasks are prevalent in competitive exams such as JEE, offering enjoyable and straightforward problem-solving experiences. This article will teach mathematical reasoning and explore methods to tackle these questions with ease and proficiency.

2.0Mathematically Acceptable Statements

The provided statement, "The sum of two prime numbers is always even," is subject to ambiguity as it could either be true or false. This ambiguity arises from the fact that the sum of two prime numbers can result in either an even or an odd number. Mathematically acceptable statements must be true or false, but not simultaneously. Hence, a clear and unambiguous statement is a fundamental prerequisite for mathematical reasoning. This forms the essence of the definition of a mathematical statement.

3.0Types of Reasoning in Mathematics

In the realm of mathematics, reasoning primarily encompasses two major types:

1. Inductive Reasoning
2. Deductive Reasoning

While other forms of reasoning, such as intuition, counterfactual thinking, critical thinking, backwards induction, and abductive induction, play roles in decision-making processes, it is inductive and deductive reasoning that predominantly characterize mathematical reasoning. These two types of reasoning will be further elaborated upon below.

Note: Inductive reasoning involves non-rigorous logical inference, where statements are generalized based on observed patterns. In contrast, deductive reasoning employs rigorous logical deduction, where statements are deemed true if the assumptions leading to the deduction are true. In the realm of mathematics, deductive reasoning holds greater significance than inductive reasoning due to its precision and reliance on established truths.

4.0Statements

In the realm of mathematics, a statement is a declarative sentence with a truth value, either true or false, but not both simultaneously.  These statements form the basis of mathematical reasoning and are essential for constructing logical arguments and proofs. Examples of mathematical statements include "2 + 2 = 4," "Every prime number greater than 2 is odd," and "The square root of 9 is 3."

Simple Statement

A simple statement in mathematics is a declarative sentence that expresses a single fact and is either true or false. It does not contain any logical connectives such as "and" "or," or "if-then." For example, "5 is an odd number" and " The result of adding 5 and 3 is 8" are both simple statements.

Compound Statement

A compound statement in mathematics is formed by combining two or more simple statements using logical connectives such as "and" "or" or "if-then." The truthfulness of a compound statement relies on the truth values of its component’s simple statements and the logical connectives used. For example, "It is raining, and the sun is shining" and "If it is raining, then I will bring an umbrella" are both compound statements.

5.0Basic Logical Connectives 

The basic logical connectives in mathematics include:

1. Conjunction (AND): Denoted by the symbol "∧" or sometimes "AND". This connective is true only when both of its component statements are true. For example, if P represents "It is sunny" and Q represents "It is warm," then P ∧ Q is true only when it is both sunny and warm.

2. Disjunction (OR): Denoted by the symbol "∨" or sometimes "OR". This connective is true if at least one of its component statements is true. For example, if P represents "It is raining" and Q represents "It is snowing," then P ∨ Q is true if it is either raining or snowing (or both).

3. Negation (NOT): Denoted by the symbol "~" or sometimes "NOT," this connective is used to form the negation of a statement. It reverses the truth value of the statement. For example, if P represents "It is cloudy," then ~P represents "It is not cloudy."

These basic logical connectives are used to form compound statements and are fundamental to mathematical reasoning and logic.

6.0Negation of Compound Statements

The negation of a compound statement is formed by negating each of its component simple statements and switching the logical connective. Here are some examples:

1. Negation of a Conjunction (AND): 

If the compound statement is P ∧ Q, then its negation is ~ (P ∧ Q), which is equivalent to (~P ∨ ~ Q).

2. Negation of a Disjunction (OR): 

If the compound statement is P ∨ Q, then its negation is ~ (P ∨ Q), which is equivalent to (~ P ∧ ~ Q).

3. Negation of an Implication (IF-THEN):

If the compound statement is P → Q, then its negation is ~ (P → Q), which is equivalent to (P ∧ ~ Q).

4. Negation of a Biconditional (IF AND ONLY IF): 

If the compound statement is P ↔ Q, then its negation is ~(P ↔ Q), which is equivalent to (P ∧ ~ Q) ∨ (~ P ∧ Q).

These rules for negating compound statements are fundamental in logic and are used extensively in mathematical reasoning.

7.0Truth Table

A truth table is a tabular representation of the possible truth values of a compound statement based on the truth values of its component propositions. Here's an example truth table for the logical connective "AND" (denoted by ∧):

In this truth table:

• P and Q represent the truth values of two propositions.
•  P ∧ Q represents the truth value of the compound statement "P AND Q."

Each row in the truth table corresponds to a different combination of truth values for P and Q. The truth value of P ∧ Q in each row is determined based on the truth values of P and Q according to the logical connective "AND."

Similar truth tables can be constructed for other logical connectives such as "OR" (denoted by ∨), "NOT" (denoted by ~), "IMPLIES" (denoted by →), and "IF AND ONLY IF" (denoted by ↔), as well as for compound statements involving multiple propositions. 

8.0Converse, Inverse and Contrapositive of the Conditional

(i) Converse: Converse of (p ⟶ q) is (q ⟶ p)

(ii) Inverse: Inverse of (p ⟶ q) is (~p ⟶ ~q)

(iii) Contrapositive: Contrapositive of (p ⟶ q) is (~q ⟶ ~p)

9.0Mathematical Reasoning Solved Questions

Example 1: Find the truth value of the statement “2 divides 4 and 3 + 7 = 8”

Solution:

2 divides 4 is true and 3 + 7 = 8 is false. so, the given statement is false.

Example 2: Find truth value of compound statement “All natural numbers are even or odd”

Solution:

p: all natural numbers are even, q: all natural numbers are odd.

Here compound statement (p ∨ q) is exclusive OR

Truth value of p is false and the truth value of q is also false.

So, truth value of compound statement (p ∨ q) is false.

Example 3: Find the truth values of (p ⟷ ~q) ⟷ (q ⟶ p).

Solution:

Example 4: Write negation of following statements:

1. "All cats scratch" 
2.  “5 is a rational number”.

Solution:

1. Some cats do not scratch.

OR

There exists a cat which does not scratch.

OR

At least one cat does not scratch.

2. 5 is an irrational number.

Example 5: Write the negation of the following compound statements:

1. All the students completed their homework and the teacher was present.
2. Square of an integer is positive or negative.
3. If my car is not in the workshop then I can go to college.

Solution:

(i)  The component statements of the given statement are:

       p: All the students completed their homework.

      q: The teacher was present.

The given statement is p and q. so its negation is ~ p or ~ q = Some of the students did not complete their homework or the teacher was not present.

(ii) The component statement of the given statements are:

p: Square of an integer is positive.

q: Square of an integer is negative.

The given statement is p or q so its negation is ~p and ~q = There exists an integer whose square is neither positive nor negative.

(iii) Consider the following statements:

        p: My car is not in the workshop. 

        q: I can go to college.

The given statement in symbolic form is p ⟶ q

Now, ~ (p ⟶ q) = p ∧ (~q)

⇒ ~ (p ⟶ q): My car is not in workshop, and I cannot go to college.

Hence the negation of the given statements is “My car is not in the workshop and I cannot go to college”.

Example 6: Write Contrapositive of statements:

(i) "If it is raining then I will not come"

(ii) "If x = 5 and y = –2 then x-2y=9"

(iii) "If two number are not equal then their square is not equal"

Solution:

(i) “If I will come then it is not raining.”

(ii) If x – 2y ≠ 9 then x ≠ 5 or y ≠ –2

(iii) “If the square of two numbers is equal then numbers are equal.”