Maharashtra Board Class 12 Maths Part 1 Chapter 1 Mathematical Logic 1.5 Solutions

Question 1. Use qualifiers to convert each of the following open sentences defined on N, into a true statement:
(i) \( x^2 + 3x - 10 = 0 \)
(ii) \( 3x - 4 < 9 \)
Answer:
(i) \( \exists x \in N \), such that \( x^2 + 3x - 10 = 0 \) is a true statement. This is because \( x = 2 \in N \) satisfies the equation \( x^2 + 3x - 10 = 0 \).
(ii) \( \exists x \in N \), such that \( 3x - 4 < 9 \) is a true statement. This is because the values \( x = 1, 2, 3, 4 \in N \) satisfy the inequality \( 3x - 4 < 9 \). Quantifiers help us define the exact range of values for which an open sentence becomes a valid mathematical statement.
In simple words: An open sentence is like a sentence with a blank space; by using the symbol \( \exists \) (which means 'there exists'), we show that there is at least one natural number that makes the math statement correct.

🎯 Exam Tip: Always state the specific values of \( x \) that satisfy the condition to clearly justify why you chose the existential quantifier \( \exists \).

Question 1. Use quantifiers to convert each of the following open sentences defined on N into a true statement:
(ii) \( 3x - 4 < 9 \)
(iii) \( n^2 \ge 1 \)
(iv) \( 2n - 1 = 5 \)
(v) \( y + 4 > 6 \)
(vi) \( 3y - 2 \le 9 \)
Answer:
(ii) \( \exists x \in N \), such that \( 3x - 4 < 9 \) is a true statement. (\( x = 1, 2, 3, 4 \in N \) satisfy \( 3x - 4 < 9 \))
(iii) \( \forall n \in N, n^2 \ge 1 \) is a true statement. (All \( n \in N \) satisfy \( n^2 \ge 1 \))
(iv) \( \exists x \in N \), such that \( 2n - 1 = 5 \) is a true statement. (\( n = 3 \in N \) satisfy \( 2n - 1 = 5 \))
(v) \( \exists y \in N \), such that \( y + 4 > 6 \) is a true statement. (\( y = 3, 4, 5, \dots \in N \) satisfy \( y + 4 > 6 \))
(vi) \( \exists y \in N \), such that \( 2y \le 9 \) is a true statement. (\( y = 1, 2, 3 \in N \) satisfy \( 3y - 2 \le 9 \))
In simple words: We use quantifiers like "for all" (\( \forall \)) or "there exists" (\( \exists \)) to turn open mathematical sentences into definite true statements depending on whether they work for all numbers or just some.

🎯 Exam Tip: Always check if the statement holds for all elements in the given set (use \( \forall \)) or at least one element (use \( \exists \)) to make it true.

Question 2. If \( B = \{2, 3, 5, 6, 7\} \), determine the truth value of each of the following:
(i) \( \forall x \in B \), \( x \) is a prime number.
(ii) \( \exists n \in B \), such that \( n + 6 > 12 \).
Answer:
(i) \( x = 6 \in B \) does not satisfy "\( x \) is a prime number". So, the given statement is false, hence its truth value is F.
(ii) Clearly \( n = 7 \in B \) satisfies \( n + 6 > 12 \). So, the given statement is true, hence its truth value is T.
In simple words: For the first statement, since 6 is in the set but is not a prime number, the "for all" statement is false. For the second statement, since we can find at least one number like 7 that makes the equation true, the "there exists" statement is true.

🎯 Exam Tip: To disprove a universal quantifier (\( \forall \)), you only need to find a single counterexample, whereas to prove an existential quantifier (\( \exists \)), finding one working example is sufficient.

Question 1. State the truth value of the following statements:
(iii) \( \exists n \in B \), such that \( 2n + 2 < 4 \).
(iv) \( \forall y \in B \), \( y^2 \) is negative.
(v) \( \forall y \in B \), \( (y - 5) \in N \).
Answer:
(iii) No element \( n \in B \) satisfy \( 2n + 2 < 4 \). So, the given statement is false, hence its truth value is F.
(iv) No element \( y \in B \) satisfy \( y^2 \) is negative. So, the given statement is false, hence its truth value is F.
(v) \( y = 2 \in B \), \( y = 3 \in B \) and \( y = 5 \in B \) do not satisfy \( (y - 5) \in N \). This is because natural numbers only include positive integers starting from 1. So, the given statement is false, hence its truth value is F.
In simple words: For statement (iii), there is no number in set B that makes the equation less than 4. For (iv), squaring any real number always gives a positive result or zero, never a negative one. For (v), subtracting 5 from numbers like 2, 3, or 5 results in zero or negative numbers, which are not natural numbers.

🎯 Exam Tip: Remember that the symbol \( \forall \) means 'for all' (every single element must satisfy the condition), while \( \exists \) means 'there exists' (at least one element must satisfy the condition).