**Mathematical Reasoning**

**Mathematical Reasoning – Meaning**

The word “Reasoning” means critical thinking, analysis and conclusion based thereon. Mathematical Reasoning means application of such analytical skill in the area of Mathematics. Mathematical Reasoning enables us to determine the Truth value of various types of Mathematical Statements, applying Mathematical Rules.

**Mathematical Statement**

In our daily life we use different types of sentences like Assertive (which declares or asserts a statement), Interrogative, Exclamatory, Imperative, Optative etc. Among them only assertive sentences are called Mathematical statement. However, all assertive sentences are not Mathematical Statements.

**Negation of Mathematical Statement**

Negation of a Mathematical Statement implies the denial (or contradiction) of the statement.

**Negation of Mathematical Statement – Examples**

**Mathematical Statement Constructs**

**Mathematical Simple Statement:**If the truth value of a statement does not depend on any other statement, then the statement is called a simple Mathematical statement. Simple statement cannot be subdivided into two or more simple statements.**Mathematical Compound Statement:**A compound statement is a combination of two or more simple statements connected by the words “and”, “or”, etc. A compound statement can be subdivided into two or more simple statements.

**Mathematical Statement Examples**

**Mathematical Simple Statement – Example**: The followings are simple statements and they cannot be sub-divided into simpler statements.

Earth moves round the sun. (ii) Sun is a star. (iv) 40 is multiple of 4

**Mathematical Compound Statement – Example**: The followings are compound statements. Each compound statement can be sub-divided into two (or more) simple statements.

**Mathematical Statement Connectives**

Connectives are words used to connect simple statements to form Compound Statements (like “and”, “or”, “if-then”, “only if”, “if and only if”, etc.).

**Compound Statement with connective ‘AND’**

Any two simple statements can be combined by using the word “and” to form compound statements.

- If each simple mathematical statement belonging to a compound mathematical statements are true then the compound mathematical statement would be True.
- If any of the simple statement connected is false, then the compound mathematical statement would be False.

**Compound Mathematical Statement**** ****(with AND) – Examples**

**Compound Mathematical Statement (with AND)– Example 1 :**Compound Statement (R): Delhi is a big city and it is the Capital of India.- R is a compound mathematical statement, formed by connecting two simple mathematical statements P & Q, using the connective “and” where
- P : Delhi is a big city – is the first constituent Simple Statement, which is TRUE
- Q: Delhi is the capital of India – is the second constituent Simple Statement, which is TRUE

Here both P and Q are true. So the value of the compound mathematical statement is “True”.

**Compound Mathematical Statement (with AND) – Example 2**:- R : 41 is a prime number and it is an even number. Here R is a compound mathematical statement and is formed by connecting two simple mathematical statements P and Q using the connective “and” where
- P : 41 is a prime number – is the first constituent Simple Statement, which is TRUE
- Q : 41 is an even number – is the second constituent Simple Statement, which is FALSE

Here P is TRUE but Q is FALSE. So the value of R is FALSE.

**Compound Mathematical Statement (with AND) – Example 3**- R : 39 is a prime number and it is an even number. Here R is a compound mathematical statement and is formed by connecting two simple mathematical statements P and Q using the connective “and” where
- P : 39 is a prime number – is the first constituent Simple Statement, which is FALSE
- Q : 39 is an even number – is the second constituent Simple Statement, which is FALSE

Here P and Q, both are FALSE. So, the value of R is FALSE.

All mathematical statements connected with “and” may not be a compound mathematical statement.

For example, “The sum of 3 and 7 is 10” is a simple mathematical statement, as there is only ONE Statement. So, it is not a compound mathematical statement.

**Compound Statement with connective ‘OR’**

Two (or more) simple mathematical statements can be combined by using “or” to form a compound

mathematical statements.

- If any one (or more) of the component simple mathematical statements of a compound mathematical statements is true then the truth value of the compound mathematical statement is True.
- If all the component simple mathematical statements are false, then the truth value of the compound mathematical statement is “false”.

**Compound Statement with connective ‘OR’ – Examples**

**Compound Mathematical Statement (with OR) – Example****1**: p: Rhombus is a quadrilateral, is a simple mathematical statement, which is TRUE

q: Rhombus is a parallelogram, is a simple mathematical statement, which is TRUE

r: Rhombus is a quadrilateral or a parallelogram, is a Compound mathematical statement of the above two simple connected mathematical statements, connected with connective ‘’OR’’

Here p & q both are simple mathematical statements and both are true.

Since both p and q are true, the truth value of r is “True” and here “OR” is Inclusive “OR”.

**Compound Mathematical Statement (with OR) – Example 2**:

p: 95 is divisible by 7, is a simple mathematical statement, which is FALSE

q: 95 is divisible by 5, is a simple mathematical statement, which is TRUE

r: 95 is divisible by 7 or by 5, is a Compound mathematical statement, of the above two simple connected mathematical statements p & q, connected with connective ‘’OR’’. Here p & q both are simple mathematical statements, connected with connective ‘’OR’’. One is TRUE & other is FALSE. So, the truth value of r is “True”.

**Compound Mathematical Statement (with OR) – Example****3**:

P: Two given straight lines AB & CD intersect at a point, is a simple mathematical statement.

Q: Two give straight lines AB & CD are parallel, is a simple mathematical statement.

R: Two straight lines AB and CD either intersect at a point or they are parallel, is a Compound mathematical statement of the above two simple connected mathematical statements, connected with connective ‘’OR’’

If P is TRUE, then Q is FALSE or if P is FALSE, then Q is TRUE, but P and Q cannot be both TRUE or cannot be both FALSE. Only one of P and Q is TRUE.

**Mathematical Statement with Inclusive & Exclusive OR**

**Mathematical Statement with Inclusive OR :**If all the component simple mathematical statements of a compound mathematical statement formed by using the connective “or” are true, then the “or” is called*Inclusive “or”.***Mathematical Statement with Exclusive OR :**In Compound mathematical Statement, if one is true and other is false, then the “or” used in compound mathematical statement is called*Exclusive “or”.*

This may be illustrated with following Example.

In a course, there are 2 subjects, Mathematics & Economics.

- One option is “You can take up Economics OR Mathematics OR both” – this is instance of “Inclusive OR”
- Another option is “ You can take up Economics OR Mathematics but not both “this is instance of “Exclusive OR”

Unless specifically stated, Inclusive *OR* is presumed. Exclusive OR is seldom used.

**Compound Mathematical Statement (with Inclusive & Exclusive OR) – Examples**

**– Example -1**: ‘*A**Triangle**can be defined as a**polygon**with three**sides**or as a polygon with three**vertices**’*is a compound mathematical statement with Inclusive “OR”. Either or both statements can be true.**Compound Mathematical Statement (with Exclusive OR – Example -2**: ‘*Today I can go to school or I can stay home’*is a compound mathematical statement with exclusive “OR”. Either option may be true but not both.**Compound Mathematical Statement (with Exclusive OR) – Example****-3**: ‘*A Student can take Urdu or Sanskrit as third language’*, is a compound mathematical statement with exclusive “OR”. A student can take either Sanskrit or Urdu, but not both.

**Mathematical Statement – Implications**

A compound mathematical statement is formed connecting two simple mathematical statements using

the connecting words “IF – THEN”, “ONLY IF” and “IF AND ONLY IF”. These connecting words are called Implications.

**Rules of Truth Value of Implications** :

**Compound Statement with connective** **IF – THEN**

**Compound Statement with IF – THEN – Example 1** : ‘’If a number is divisible by 6, then it must be divisible by 3″. It is compound mathematical statement.

- p: A number is divisible by 6
- q: The number is divisible by 3.

Here if p is true, then q must be true. But if p is not true (the number is not divisible by 6) then we cannot say that it is not divisible by 3 (e.g. 123 is not divisible by 6 but it is divisible by 3).

**Compound Statement with IF – THEN – Example 2 **: “If 42 is divisible by 7, then sum of the digits of 42 is divisible by 7”, is a compound mathematical statement.

- p: 42 is divisible by 7 is a is a simple mathematical statement, which is TRUE.
- q: The sum of the digits of 42 is divisible by 7, is a simple mathematical statement, which is FALSE.

**Compound Statement with IF – THEN – Example 3 **: “If the person is born in India, then the person is a citizen of India”, is a compound mathematical statement.

- p: The person is born in India.
- q: The person is a citizen of India.

Here, if p is TRUE, then q is TRUE.

**Compound Statement with IF – THEN – Example** 4: If it rains, then the class will not go out to picnic.

This means that [if it rains], the statement [the class will not go out to picnic] is TRUE. However, If [the class will not go out to picnic] does not mean that [It is raining]

**Contrapositive and Converse Mathematical Statement**

If a compound mathematical statement is formed with two simple mathematical statements p and q

using the connective “IF-THEN”, then the *Contrapositive* and *Converse* statements of compound statement can also be formed.

**Contrapositive Mathematical Statement**

**Converse Mathematical Statement**

**Contrapositive & Converse Mathematical Statement** – **Example 1**: “If a number is divisible by 6, then it is divisible by 3”, is a compound mathematical statement.

- Its contrapositive statement is: “if a number is not divisible by 3, then it cannot be divisible by 6”.
- Its Converse statement is : “If a number is divisible by 3, then it is divisible by 6”.

**Contrapositive & Converse Mathematical Statement** – **Example 2 **: “If it is raining then the grass is wet”, is a compound mathematical statement.

- The contrapositive is “If the grass is not wet, then it is not raining.”
- The converse is “If the grass is wet then it is raining.”

**Compound Mathematical Statement with connective** “**ONLY IF”**

The compound mathematical statement formed with two simple mathematical statements, p and q, using the connective word “ONLY IF” implies that p happens only if q happens.

**Compound Mathematical Statement with connective**“**ONLY IF” – Example 1.**The compound mathematical statement ‘Samir will attend the party only if Kanti will be there’ means that ‘Samir will attend the party’ will happen (i.e becomes TRUE) only if Kanti will be at the Party’ happens (i.e becomes TRUE).**Compound Mathematical Statement with connective**“**ONLY IF” – Example 2.**The compound mathematical statement ”The triangle ABC, will be equilateral only if AB = BC = CA.” formed from two simple statements i) p: ‘The triangle ABC is equilateral’, ii) q: ‘In the triangle ABC, AB = BC = CA’ means that the statement p: ‘The triangle ABC, will be equilateral’, is TRUE only if the statement q:’ If AB = BC = CA’ is TRUE.

**Compound Mathematical Statement with **‘’**IF AND ONLY IF (IFF)’’**

A Biconditional compound mathematical is formed with two simple mathematical statements using the connecting words “if and only if”.

**Compound Mathematical Statement with **‘’**IF AND ONLY IF – Example 1**: Two triangles are congruent IF AND ONLY IF the three sides of one triangle are equal to the three sides of the other triangle.”

This statement can be written as:

- If two triangles are congruent, then the three sides of one triangle are equal to the three sides of the other triangle.
- If the three sides of one triangle is equal of the three sides of the other triangle, then the two triangles are congruent.

Let p: Two triangles are congruent. q: Three sides of one triangle are equal to the three sides of the other triangle.

**Mathematical Statement with Quantifiers**

In some mathematical statements some phrases (called *Quantifiers*) like “There exists”, “For all” (or for every) are used,

**Mathematical Statement with**:*Exitential Quantifier*

**Mathematical Statement with***Universal Quantifier***Truth value for Statements with Quantifiers (For Numbers)****– Examples**

**Truth value for Statements with Quantifiers (For Equations) Examples**: Using Quantifiers, equations expressed into statement :

**Mathematical Statement with Contradiction**

Contradiction is a process by which we can test the truth value of the statement by evaluating the truth value of contradictory statement.