FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 1.43(iii) “There exists” is the quantifier. The truth value of the statement is “true” because for any natural number
nú 8 , the relation n 2 5−.
(iv) “For all” is the quantifier. The truth value of the statement is “True”, because x = 0 is a real number and
x^20.
Example 73 : Using Quantifiers, express the following equations into a statement :
(i) n +2 > n, n ∈ N. (ii)^2 <0x , x ∈ I– (where I– denotes the set of all negative integers) (iii) x + 1 > 3, χ∈R.
Solution :
(i) There exists a natural number n∈N such that n 2 >+ n. The statement is true. The quantifier is “There
exists”.
(ii) For all negative integers x ∈ I–, x^2 < 0. The statement is false because the square of any negative integer
is greater than zero. The quantifier is “For all”.
(iii) There exists a real number x ∈ R such that x 1 >+ 3. The statement is true because for all real number
x>2, the relation x+1>3 is true. The quantifier is “There exists”.
Contradiction :
Contradiction is process by which we can test the validity of a given statement.
Let P : “If n>4, then^2 >16n , where n is any real number”. We shall show that P is true by contradiction
process as follows :
Let n be a real number and n> 4 but x^2 (16.
Q n^2 (16, ∴^2 ≤16n. or,^2 ≤− 016n (n–4) (n+4) ≤ 0 ..... (1)Since n > 4, n≠ 4 or n 4 ≠− 0 and n 4 >− 0.
So from (1) we get n 4 ≤+ 0 or n −≤ 4. It is not possible because n > 4. So our assumption must be wrong
i.e., n^2 (16 is wrong.
(^2) >∴ 16n.
Self Examination Questions
- Examine whether the following sentences are mathematical statements or not (give reasons) :
(i) The sun is a star (ii) Go to the market. (iii) Who is the chief-minister of West Bengal? (iv) The prime
factors of 15 are 3 and 5. (v) May God bless you! (vi) How nice the building is! (viii) x^2 –x+6=0
(viii) The roofs of 2x^2 –3x–5 = 0 are 1 and − 25.
Ans. (i) Mathematical Statement (ii) No (iii) No (iv) Mathematical statement (v) No (vi) No (vii) No (viii)
Mathematical statements
- Write the truth value of each of the following sentences and comment whether a mathematical
statement or not :
(i) Tomorrow is Wednesday. (ii) Every rectangle is a square (iii) 2 is an irrational number (iv) Alas! I am
undone. (v) There are 31 days in the month of July and August in each year (vi) Mathematics is an
interesting subject.