OP Malhotra Class 11 Maths Solutions Chapter 27 Mathematical Reasoning Exercise 27 (B)

S Chand Class 11 ICSE Maths Solutions Chapter 27 Mathematical Reasoning Ex 27(b)

Question 1. Identify the quantifier in the following statements.
(i) There exists a capital city for every state of India.
(ii) For every real numbers, x is less than x + 1.
(iii) At least one natural number is not a prime number.
Answer:
(i) The quantifier in this statement is "There exists". This shows that at least one such city can be found.
(ii) The quantifier here is "For every". This means the statement holds true for all real numbers.
(iii) The quantifier in this statement is "At least one". This indicates that one or more such natural numbers exist. Quantifiers help specify the number of elements for which a statement is true.
In simple words: A quantifier tells us how many things a statement is about. It could be "there exists one" or "for every one".

🎯 Exam Tip: Quantifiers like "for all", "for every", "there exists", and "at least one" are key to understanding the scope of a mathematical statement. Identifying them correctly is crucial for logical reasoning.

Question 2. Symbolise the following statements.
(i) There is at least one number in the set of natural numbers which is equal to 'its' cube.
(ii) The square of every real number is positive.
(iii) There exists at least one number in A = {5, 7, 8, 9, 10} which is an even number.
(iv) For every real number x, x < x + 1.
(v) The square roots of all prime numbers are irrational numbers. (Let P denote the set of prime numbers and Q that of irrational numbers).
Answer:
(i) \( \exists x \in N \) such that \( x = x^3 \). This means there is at least one natural number whose cube is itself.
(ii) \( \forall x \in R, x^2 > 0 \). This symbolises that for all real numbers, their square is greater than zero.
(iii) \( \exists x \in A \) such that \( x \) is even. Here, A is the set {5, 7, 8, 9, 10}, and we are looking for at least one even number in it.
(iv) \( \forall x \in R, x < x + 1 \). This statement is true for all real numbers, meaning any real number is always less than itself plus one.
(v) \( \forall p \in P, \sqrt{p} \in Q \). This means for every prime number \( p \), its square root belongs to the set of irrational numbers. Symbolizing statements helps to represent them clearly and precisely in mathematical logic.
In simple words: We change the word statements into math symbols. For "there is at least one", we use \( \exists \). For "for every", we use \( \forall \). We also use symbols like \( \in \) for 'is a member of' and \( = \) for 'equals'.

🎯 Exam Tip: When symbolizing statements, pay close attention to the quantifiers ("for every", "there exists") and the sets involved (natural numbers, real numbers, prime numbers) to choose the correct symbols and express the logical meaning precisely.