Chapter 14: Mathematical Reasoning

Chapter 14

Mathematical Reasoning

Statements and Connecting words:

Statements in Mathematics:

Definition:

It is the science of reaching some conclusions on the basis of some statements, or it is the science or method of correct reasoning.

Terminology:

Statement:

 A statement (or preposition) is an assertion which is either true or false but not the both.

The truth values of a statement is represented by the letter ‘T’ or ‘F’.

Examples of statements:

1. The earth is round. (T)
2. Delhi is the capital of India. (F)
3. 1 + 2 = 3 (T)
4. A triangle has three sides. (T)
5.  
6. Two is greater than five (F)
7. Sun rises in west (F)

Examples of not statements.

1. Are you going to Delhi tonight?
2. Oh God ! I am so sorry.
3. May God grant you long life.
4. Bring a glass of water.
5. How beautiful is that flower.

The statements like:

X is less than 10.

u is the father of v.

are not statements as they contains variables.

FUZZY PROPOSITION:

Definition: The statements whose truth value cannot be determined in the absence of any measuring yardstick are called fuzzy proposition.

The sentences like,

Mohan is a wise man.

Harish is very rich.

Sangram is a naughty boy etc.

are the sentences which are neither true or false cannot be determined in the absence of any measuring yardstick. These type of sentences are called fuzzy propositions.

Statements are normally represented by small Roman letters like p, q, r, s, t, etc.

CONNECTIVES STATEMENTS:

Simple statement:

A single statement whose truth value does not depend upon any other statement is called simple statement.

Example: Bhubaneswar is the capital of Orissa.

India became independent in 1947.

Compound statement:

Compound statement is a combination of two or more simple statements.

Example: Either he is intelligent or he is working.

If you work hard, you can be selected in IIT.

(i) Negation (~):

If a proposition ‘p’ is modified by the word ‘not’ a new statement results. It is called negation. p and it is represented by ~ p.

Example:

p: Bhagat Singh was a great patriot.

~ p: Bhagat Singh was not a great patriot.

OR

The denial of a statement is called the negation of the statement. In other words, if p is a statement, then the negation of p is also a statement and is denoted by ∼ p, and read as ‘not p’.

While forming the negation of a statement, phrases like, “It is not the case” or “It is false that” are also used.

Let us consider the statement:

p: Bhubaneswar  is a city.

The negation of this statement is

~p: It is not the case that Bhubaneswar is a city

This can also be written as

~p: It is false that Bhubaneswar is a city.

This can simply be expressed as

~p: Bhubaneswar is not a city

These are the different ways to write the negation of a given sentence.

Axiom of negation:

For any proposition p, if p is true, then ~ p is false and if p is false, then ~ p is true.

(ii) Conjunction (Ù)

            If p and q be two statements, then p Ù q means ‘both p an q’.

            This type of statement is called conjunction.

Example:

            p : Rama plays well.

            q : Rama is a good student.

            p Ù q : Rama plays well and be is a good student.

Example :

p: 30 is a multiple of 5, 6 and 7.

This statement has the following component statements

q: 30 is a multiple of 5.

r: 30 is a multiple of 6.

s: 30 is a multiple of 7.

Here, we know that the first is false while the other two are true.

Axiom of conjunction:

A conjunction p Ù q is true if both p and q are true and false if either p or q is false or both false.

 The conjunction table

(iii) Disjunction (Ú):

If p and q be two statements. The p Ú q means ‘either p or q or both’. These type of statements are called disjunction.

Example:

            p : you go to market.

            q : you bring a pen.

            p Ú q : either you go to market or you bring a pen.

Example:

Compound statements can also be written using the word “Or”. However, in this case, we can determine whether an inclusive “Or” or exclusive “Or” is used.

Let us consider the following examples to understand the difference between inclusive and exclusive Or.

Example 1: The school is closed if it is a holiday or a Sunday.

Here also “Or” is inclusive since school is closed on holidays as well as on Sunday.

Example 2: Two lines intersect at a point or are parallel.

Here “Or” is exclusive because two lines cannot intersect and parallel together.

It is essential to note the difference between these two ways because we require this when we check whether the statement is true.

Understanding of conditions

Rules for the compound statement with ‘Or’

• A compound statement with an ‘Or’ is true when one component statement is true or both are true.
• A compound statement with an ‘Or’ is false when both the component statements are false.

Quantifiers

Quantifiers are phrases like “There exists” and “For all”. A word closely connected with “there exists” is “for every” (or for all). Hence, the words “And” and “Or” are called connectives and “There exists” and “For all” are called quantifiers.

Axiom of disjunction:

If p, q are propositions, then p Ú q is true if p is true or q is true or both true, and it is false if both p & q are false.

(iv) Conditional or implication (Þ):

If p and q be two statements, then p Þ q means ‘p implies q’. This type of statements are called conditional statements.

Example:

            p : x² = 4

            q : x = ± 2

Axiom of conditional:

A conditional p Þ q is false if p is true and q is false, otherwise true.

Here p is called hypothesis and q is called conclusion.

(v) Biconditional (Û):

If p and q are two statements, then p Û q means ‘p if and only if q’. This type of statements are called Biconditional statements.

Example:

            p : Five is a prime number.

            q : It has no proper divisor.

            P Û q : Five is prime if and only if it has no proper divisor

 

Truth Table:

The tabular classification of complex statements given above are taken as axioms upon which logical decissions are based and such tables are called truth tables.

Given a conditional p ® q, three other propositions related to itcanbe framed.

1. Converse : q ® p
2. Inverse : ~ p ® ~ q
3. Contra positive : ~ q  ® ~ p.

 

Fallacy:           A statement which is always false is called fallacy.

Tautology:      A statement which is always true is called tautology.

Equivalent:    Two statements are equivalent if their truth values are identical i.e. entries in the two columns are exactly same.

Example:         p  ~ (~p) are equivalent.

Some important tautologies are given below.

(i) Law of addition:

If p & q are two statements, then

p Þ p Ú q,   q Þ p Ú q

(ii) Law of Implication: 

 p Ù q Þ p,   p Ù q Þ = q, where p & q are two statements.

(iii) Commutative laws:

If p & q are two statements, then p Ù q Û q Ù q, p Ú q Û q Ú p

(iv) Associative laws:

If p, q & r are two statements, then

p Ù  (q Ù r) Û (p Ù q) Ù r

p Ú (q Ú r) Û (p Ú q) Ú r

(v) Distributive laws:

If p, q and r are statements, then

1. 1. pÙ(qÚr)Û(pÙq) Ú (p Ù r)
   2. pÚ(q Ùr)Û(pÚq) Ù (p Ú r)

(vi) Idempotent Laws:

If p is a statement, then

p Ú p Û p, p Ù p Û p

(vii) Law of double negation:

If p is a statement, then p Û ~ (~p)

(viii) Law of contra positive:

If p, q are two statements, then (p Þ q) Û (~ q Þ ~ p)

(ix) Law of inclusive middle:

If p is a statement, then p Ú ~ p

(x) De-morgans laws:

If p & q are two statements, then

1. ~  (p Ù q) Û (~ p Ú ~ q)
2. ~ (p Ú q) Û ~ p Ù ~ q

These are familiar examples of tautologies.

Compound statements

Many mathematical statements are obtained by combining one or more statements using some connecting words like “and”, “or”, etc.

Now, suppose two statements are given as below:

p: 5 is an odd number.

q: 5 is a prime number.

We can combine these two statements with “and”.

r: 5 is both an odd and prime number.

This is a compound statement.

From this, we can write the following definition:

Examples

 

Question: Prove that ~ (pÙq) = ~ p Ú ~ q by constructing a truth table.

Solution:

This can be proved by constructing the following truth table.

 Since the truth values of column 6 & 7 are same it proves that

            ~ (pÙq) = ~ p Ú ~ q

Question: Prove that {(p®q) Ù (q ® r)} ® (p ® r) is a tautology by constructing the corresponding truth table.

Solution:

Question: If r is any false statement then show that p ®r  is logically equivalent to p Ù (~ q) ® r

Solution:

Here we observe that the truth values of column 4 and column 7 are equal in the above truth table, it proves that p ®q is logically equivalent to

p Ù (~ q) ® r.

Solution:

Construct the truth table of [p ® (pÚq)]  ® [q  ® (pÙq)]

Hence or otherwise decide, we either it is a tautology or not.

Question: Examine whether  ~ q Ú p and q ® p are equivalent.

Solution:

Let’s form a truth table of ~ q Ú p  and q ® p to decide weather they are equivalent or not.

As the truth values of column 4 & 5 are same, hence they are equivalent.

Question: Constructing the truth table, state whether the following statement is a tautology  not.

 (pÙq) Ú (~p Ú ~ q)

As all the truth values of column 7, are true. Hence the given statement is a tautology.

Question: Draw the truth table of the following statement.

~ (p® q) « [(~ p Ú q) « ~ q]  

Or

Contrapositive and converse

Contrapositive and converse are certain other statements which can be formed from a given statement with “if-then”.

• The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒ ∼p .
• The converse of a statement p ⇒ q is the statement q ⇒ p.
• p ⇒ q together with its converse, gives p if and only if q

Solution:

Validating the statements involving the connecting words and statements:

METHODS OF PROOF

Mathematical proofs are deductive in nature. A mathematical argument consists of a set of propositions (called hypotheses) and a set of conclusions. It is called logically valid if the conclusions are true. Otherwise called logically invalid.

There are several methods to show an argument is valid which are:

1. Principle of syllogism.
2. Principle of reduction absurdum.
3. Law of contrapositive.
4. Principle of mathematical induction.

OR

Validating Statements

The following methods are used to check the validity of statements:

(i) direct method

(ii) contrapositive method

(iii) method of contradiction

(iv) using a counter example.

Difference between contradiction, contrapositive, and converse:

Contrapositive and converse

Contrapositive and converse are certain other statements which can be formed from a given statement with “if-then”.

• The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒ ∼p .
• The converse of a statement p ⇒ q is the statement q ⇒ p.
• p ⇒ q together with its converse, gives p if and only if q