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.