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

Question 1. Write each sentence in the "If .......... then" form.
(i) Roses are vegetables if carrots are flowers.
(ii) All ducks are birds.
(iii) Vertical angles are equal.
(iv) Freezing water expands.
(vi) A racer wins the race only if he runs fast.
(vii) Any two parallel lines are coplanar.

Answer: This type of conversion helps us clearly see the conditional relationship between two statements. The "if" part is the condition, and the "then" part is the result.
(i) If carrots are flowers then roses are vegetables.
(ii) If the creature is a duck then it is a bird.
(iii) If two angles are vertical angles then they are equal.
(iv) If the water is freezing then it expands.
(v) If the set has no element then it is called the empty set.
(vi) If the racer runs fast then the racer wins the race.
(vii) If two lines are parallel then they are coplanar.
In simple words: We take each sentence and change its wording to clearly show "if this happens, then that happens". This makes the cause and effect easy to understand.

🎯 Exam Tip: When converting sentences, correctly identify the cause (the 'if' part) and the effect (the 'then' part) to ensure accurate transformation.

Question 2. Let p be "I will marry her," and let q be “she is beautiful.” Translate into symbolic form.
(i) If she is beautiful, then I will marry her.
(ii) If I will marry her, then she is beautiful.
(iii) If she is beautiful, then I will not marry her.
(iv) If she is not beautiful, then I will not marry her.
(v) If I will not marry her, then she is not beautiful.
(vi) If she is beautiful, then I will not marry her.

Answer: We use logical symbols to represent these statements more simply. Here, \( p \) stands for "I will marry her" and \( q \) stands for "she is beautiful". The arrow \( \implies \) means "if...then...", and the tilde \( \sim \) means "not".
(i) \( q \implies p \)
(ii) \( p \implies q \)
(iii) \( q \implies p \)
(iv) \( \sim q \implies \sim p \)
(v) \( \sim p \implies \sim q \)
(vi) \( q \implies \sim p \)
In simple words: We use special symbols to write these sentences about marrying and beauty. \( p \) means "I will marry her", and \( q \) means "she is beautiful". The arrow means "if...then...", and the wavy line means "not".

🎯 Exam Tip: Always clearly define what \( p \) and \( q \) represent. Pay close attention to words like "not" or "only if" as they change the logical relationship and require careful use of \( \sim \) and correct arrow direction.

Question 3. Determine whether p, q and ‘If p, then q' are true or false in each case given below :

Answer: For each statement, we first determine the truth value of 'p' and 'q' individually. Then, we find the truth value of the conditional statement 'If p, then q'. This combined statement is only false if 'p' is true and 'q' is false; otherwise, it is true. Understanding truth tables is fundamental for building sound logical arguments.
(i) Given p: 3 is a prime number (True)
q : 3 is an even number (False)
Then \( p \implies q \) is false
(ii) p : 5 < 7 (True)
q : 5 is odd (True)
Then \( p \implies q \) is true
(iii) p : 3 > 2 (True)
q : 2 × 7 = 14 (True)
Then \( p \implies q \) is true.
(iv) p : 1 > 4 (False)
q : 2 is even (True)
Then \( p \implies q \) is true
(v) p : 5 × 3 = 16 (False)
q : 2 + 7 = 6 (False)
Then \( p \implies q \) is true
(vi) p : \( 3 \left( \frac{5}{6} \right) < 1 \) (False) \( \implies 3 \times \frac{5}{6} = \frac{15}{6} = 2.5 \), which is not less than 1.
q : \( 8 - \frac{3}{6} > 9 \) (False) \( \implies 8 - 0.5 = 7.5 \), which is not greater than 9.
Then \( p \implies q \) is true
In simple words: We check if the first part (p) is true, then if the second part (q) is true. After that, we decide if the "if p, then q" rule is true or false. It is only false if "p" is true but "q" is false.

🎯 Exam Tip: Remember the key rule for conditional statements: \( p \implies q \) is FALSE only when \( p \) is TRUE and \( q \) is FALSE. In all other cases (TRUE-TRUE, FALSE-TRUE, FALSE-FALSE), the statement is TRUE.

Question 4. Write T before each true statement and write F before each false statement. Then give the truth value of the conditional expressed.
Answer: For each statement, we break it into the 'if' part (p) and the 'then' part (q). We find out if each part is true (T) or false (F). Then, we figure out if the whole 'if p, then q' statement is true or false. This statement is only false when 'p' is true and 'q' is false. Carefully identifying the truth values of individual components is essential for correctly evaluating complex logical statements.
(i) Let p : Asia is a continent (T)
q: Delhi is in Japan (F)
Then \( p \implies q \) is false (F)
(ii) Let p : Monkeys climb trees (T)
q: 6 is divisible by 2 (T)
Then given statement \( p \implies q \) is True (T)
(iii) Let p: Oxygen is a gas (T)
q: Gold is a compound (F)
Then given statement \( p \implies q \) is false (F)
(iv) Let p : water is dry (F)
q: snow is hot (F)
Then \( p \implies q \) is true (T)
(v) Let p : Snow is cold (T)
q : water is wet (T)
Then \( p \implies q \) is True (T)
(vi) Let p : 3 = 5 (F)
q: 7 is a prime number (T)
Then \( p \implies q \) is True (T)
(vii) Let p : \( 5 \times 6 - 4 = 21 \) (F) \( \implies 30 - 4 = 26 \neq 21 \)
q : \( 2 \left( \frac{5}{15} + 3 \right) = \frac{20}{3} \) (T) \( \implies 2 \left( \frac{1}{3} + 3 \right) = 2 \left( \frac{1+9}{3} \right) = 2 \left( \frac{10}{3} \right) = \frac{20}{3} \)
Then \( p \implies q \) is true (T)
(viii) Let p: triangle is a rectangle (F)
q: Circle is a rhombus (F)
Then \( p \implies q \) is true (T)
(ix) Let p : 51 is the product of + 17 and (-3) (F) \( \implies 17 \times (-3) = -51 \neq 51 \)
q: lions can fly in the air (F)
Then \( p \implies q \) is true (T)
(x) Let P: \( \sqrt{5} \) is an integer (F)
q: 3 is an integer (T)
Then \( p \implies q \) is true (T)
In simple words: For each sentence, we find the first part (p) and the second part (q). We decide if p is true or false, and if q is true or false. Then we decide if the whole "if p, then q" sentence is true or false. It is only false if the first part is true but the second part is false.

🎯 Exam Tip: Break down each complex conditional statement into its simple propositions \( p \) and \( q \). Assign their individual truth values before evaluating the truth value of the entire conditional statement \( p \implies q \).