Paper 4: Fundamentals of Business Mathematics & Statistic

1.40 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Arithmetic

Here if p is true, then q must be true. But if p is not true i.e., a number is not divisible by 6 then we cannot say

that it is not divisible by 3. For example: 123 is not divisible by 6 but it must be divisible by 3.

Here p is called “sufficient condition” for q and q is called the “necessary condition” for p.

If p is true then q is false then p e q is false.

Regarding the truth value of “if p then q” i.e., p e q (the validity of p e q) the following is kept in mind:

(i) If p is true and q is true, then the statement p e q is true.

(ii) If p is false and q is true, then the statement p e q is true.

(iii) If p is false and q is false, then the statement p e q is true.

(iv) If p is true and q is false, then the statement p e q is false.

Example 62 : “If 42 is divisible by 7, then sum of the digits of 42 is divisible by 7”. – It is a compound

mathematical statement.

p: 42 is divisible by 7. q: The sum of the digits of 42 is divisible by 7.

Here p is true but q is not true.

∴ p e q is false.

∴ The truth value of the given statement is false.

Example 63 : “If 123 is divisible by 3, then the sum of the digits of 123 is not divisible by 3”. – It is a compound

mathematical statement.

p: 123 is divisible by 3. q: the sum of the digits of 123 is not divisible by 3.

Here p is true and q is not true.

• p e q is false.


• The truth value of the given statement is false.
  Example 64 : “If anybody is born in India, then he is a citizen of India”. It is compound mathematical statement.
  p: Anybody is born in India. q: He is a citizen of India.
  Here if p is true, then q is true. So p e q is true.
  Contrapositive and converse 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.
  The contrapositive statement of “if p, then q” is “if – q, then – p” and the converse statement is “if q then p”.
  Example 65 : “If a number is divisible by 6, then it is divisible by 3”. – It is a compound mathematical
  statement.
  Its contrapositive statement is
  “if a number is not divisible by 3, then it cannot be divisible by 6”.
  The Converse statement is
  “If a number is divisible by 3, then it is divisible by 6”.
  Example 66 : “If a number is an even number, then its require is even”.
  Its contrapositive statement is
  “If the require of a number is not even, then the number is not even”.