Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 1.41

The converse statement is
“If the require of a number is an even number, then the number is even”.
Example 67 : Examine the truth value of the following statement using contrapositive odd integer, then
both X and Y are positive odd integers”. (x and y are positive integer)
Solution: P : “xy is a positive odd integer, (x, y are positive integers)”.
q: “Both x and y are positive odd integers.
∴ The given statement is “If p, then q.”
Its contrapositive statement is ∼ q ⇒ i p that is q is false implies p is false.
Let q be false.
∴ q is false ⇒ x and y both are not positive integers ⇒ at least one x and y is an even positive integer.
Let x be an even positive integer and x = 2m, m is any positive integer.
∴ xy = 2my = 2 (my) which is a positive even integer.
So xy is not a positive odd integer i.e., p is not true or p is false.
∴ q is false ⇒ p is false or ~q is true ⇒ ~p is true
∴ The given statement is true.
(ii) The word “only if”:
Let p and q be two given simple mathematical statements. If a compound mathematical statement
is formed with p and q using the connective word “only if” then it implies that p only if q that is p
happens only if q happens.
Example 68 : “The triangle ABC, will be equilateral only if AB = BC = CA.”
Here, p: The triangle ABC is equilateral.
q: In the triangle ABC, AB = BC = CA.
Example 69 : “A number is an even integer only if the number is divisible by 2.”
Here, p: A number is an even integer.
q : The number is divisible by 2.

N.B: If the implication for a compound mathematical statement contains “if-then” or “only-if” then the statement

is called conditional statement. “if p, then q” – here p is called antecedent and q is called consequent.

Example 70 : Obtain the truth value of
(i) If 5 + 6 = 11, then 11 – 6 = 5.
(ii) If 5 + 8 = 12, then 12 + 8 = 20.
(iii) If 6 + 9 = 14, then 14 – 7 = 8
(iv) If 7 + 8 = 15, then 8 – 7 = 2

Solution:
(i) Since p: 5 + 6 = 11 is true and q: 11 – 6 = 5 is true, so p ⇒ q i.e., the given statement is true.

(ii) Since p:5 + 8 = 12 is false and q: 12 + 8 = 20 is true, so p ⇒ q i.e., the given statement is true.
(iii) Since p:6 + 9 = 14 is false and q: 14 – 7 = 8 is false, so p ⇒ q i.e., the given statement is true.

(iv) Since p: 7 + 8 = 15 is true and 8 – 7 = 2 is false, so p ⇒ q i.e., the given statement is false.