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

 

Question 1. Write the converse, inverse and contra-positive of the following statements.
(i) If you do not drink your milk, you will not be strong.
(ii) If you drink milk, you will be strong.
(iii) You will be strong only if you drink your milk.
Answer:
(i) Given statement: If you do not drink your milk (p), then you will not be strong (q).
Converse: \( q \implies p \): If you are not strong, then you do not drink your milk.
Inverse: \( \sim p \implies \sim q \): If you drink your milk, then you will be strong.
Contrapositive: \( \sim q \implies \sim p \): If you are strong, then you drink your milk.
(ii) Given statement: If you drink milk (p), then you will be strong (q).
Converse: If you are strong, then you drink your milk.
Inverse: If you do not drink your milk, then you are not strong.
Contrapositive: If you are not strong, then you do not drink your milk.
(iii) Given statement: You will be strong only if you drink your milk. This means: If you are strong (p), then you drink your milk (q).
Converse: \( q \implies p \): If you drink your milk, then you are strong.
Inverse: \( \sim p \implies \sim q \): If you are not strong, then you do not drink your milk.
Contrapositive: \( \sim q \implies \sim p \): If you do not drink your milk, then you are not strong.
In simple words: For each "if-then" statement, you need to change its form in three ways: converse (swap the "if" and "then" parts), inverse (add "not" to both parts), and contrapositive (swap and add "not" to both parts). Remember to apply the "not" correctly to the original conditions.

🎯 Exam Tip: When forming the contrapositive, always negate both the hypothesis and the conclusion *and* swap their positions. For "only if" statements, correctly identify which part is the condition and which is the conclusion first.

 

Question 2. Write the converse of each of the following statements. In which cases is the converse true?
(i) If an integer is even, then its square is divisible by 4.
(ii) If it is raining, then there are clouds in the sky.
(iii) In order to get this job, I must be a graduate.
(iv) If Mr. Sexena is elected to office, then all our problems are over.
Answer:
(i) Original statement (p \( \implies \) q): If an integer is even, then its square is divisible by 4.
Converse (q \( \implies \) p): If the square of an integer is divisible by 4, then the integer is even. (True)
(ii) Let p: it is raining. Let q: There are clouds in the sky.
Original statement (p \( \implies \) q): If it is raining, then there are clouds in the sky.
Converse (q \( \implies \) p): If there are clouds in the sky, then it is raining. (False)
(iii) Let p: I get this job. Let q: I must be a graduate.
Original statement (p \( \implies \) q): If I get this job, then I must be a graduate.
Converse (q \( \implies \) p): If I am a graduate, then I can get this job. (False)
(iv) Let p: Mr. Sexena is elected to office. Let q: all our problems are over.
Original statement (p \( \implies \) q): If Mr. Sexena is elected to office, then all our problems are over.
Converse (q \( \implies \) p): If all our problems are over, then Mr. Sexena is elected to office. (False)
In simple words: To find the converse, just switch the "if" part and the "then" part of the statement. Then, check if this new statement is actually true or false based on what we know. A statement like "If it rains, there are clouds" is true, but its converse "If there are clouds, it rains" is not always true because clouds can be there without rain.

🎯 Exam Tip: When evaluating the truth of a converse statement, think of counterexamples. If you can find just one situation where the converse is false, then the converse itself is false.

 

Question 3. Consider the statements : p: You will work hard q : You will become wealthy. Translate each of the symbolic statements into an English sentence.
(i) P \( \implies \) q
(ii) q \( \implies \) p
(iii) (\( \sim \)p) \( \implies \) (\( \sim \) q)
(iv) (\( \sim \)q) \( \implies \) (\( \sim \)p)
Answer:
Given statements:
p: You will work hard
q: You will become wealthy

(i) \( p \implies q \): If you will work hard, then you will become wealthy.
(ii) \( q \implies p \): If you will become wealthy, then you will work hard.
(iii) \( \sim p \implies \sim q \): If you will not work hard, then you will not become wealthy.
(iv) \( \sim q \implies \sim p \): If you will not become wealthy, then you will not work hard. This statement connects not getting rich with not working hard.
In simple words: We are just changing math symbols into everyday English sentences. The arrow \( \implies \) means "if...then". The squiggly line \( \sim \) means "not". So, we replace the letters with their phrases and the symbols with "if...then" or "not".

🎯 Exam Tip: Pay close attention to the `~` symbol, which stands for "not". Make sure to negate the correct part of the statement (either p or q) when translating to English.

 

Question 4. Compare the following statements :
(i) P \( \implies \) q
(ii) If p, then q
(iii) p is a sufficient condition for q.
(iv) q is a necessary condition for p.
(v) p, only if q.
Answer:
All the given five statements mean the same thing and are equivalent to \( p \implies q \). These are just different ways to express a conditional relationship in logic. For example, "p is a sufficient condition for q" means that p happening is enough for q to happen.
In simple words: All these different phrases are just other ways to say "if p, then q". They all describe the same basic idea of cause and effect in logic.

🎯 Exam Tip: It is crucial to understand that "p is a sufficient condition for q" and "q is a necessary condition for p" both imply the same conditional statement: \( p \implies q \).

 

Question 5. Construct truth tables for each of the following :
(i) (\( p \implies q \)) \( \wedge \) (\( q \implies p \))
(ii) \( q \implies [(\sim p) \vee q] \)
(iii) \( [(\sim p) \wedge q] \implies (p \vee q) \)
Answer:
(i) The truth table for \( (p \implies q) \wedge (q \implies p) \) is given below:

(ii) The truth table for \( q \implies [(\sim p) \vee q] \) is given below:

(iii) The truth table for \( [(\sim p) \wedge q] \implies (p \vee q) \) is given below:

In simple words: A truth table lists all possible "true" (T) or "false" (F) combinations for the basic statements (like p and q) and then shows the truth value for a more complex statement built from them. We figure out each small part step by step until we get the final column.

🎯 Exam Tip: Remember the truth conditions for each logical operator: \( \implies \) (conditional is false only if T \( \implies \) F), \( \wedge \) (conjunction is true only if both are T), \( \vee \) (disjunction is false only if both are F), and \( \sim \) (negation flips the truth value).

 

Question 6. Write the converse, inverse and contra-positive for the statement (\( \sim p \)) \( \implies \) q.
Answer:
Given statement: \( (\sim p) \implies q \)
Converse: \( q \implies (\sim p) \)
Inverse: \( \sim (\sim p) \implies \sim q \). This simplifies to \( p \implies \sim q \).
Contrapositive: \( \sim q \implies \sim (\sim p) \). This simplifies to \( \sim q \implies p \). This connects the negation of the conclusion with the original hypothesis.
In simple words: Starting with "If not p, then q", we change it around. For the converse, we swap the two parts. For the inverse, we add "not" to both parts of the original statement. For the contrapositive, we swap the parts and add "not" to both of them.

🎯 Exam Tip: When dealing with negation `~`, remember that `~(~p)` simplifies to `p`. This is a common simplification in logical statements.

 

Question 7. Write the inverse of the converse of p \( \implies \) q.
Answer:
Given statement: \( p \implies q \)
First, find its converse: \( q \implies p \)
Next, find the inverse of the converse \( (q \implies p) \). To do this, we negate both parts of the converse.
Inverse of converse: \( \sim q \implies \sim p \). This is also the contrapositive of the original statement.
In simple words: First, swap the "if" and "then" parts of the original statement. Then, add "not" to both of the swapped parts.

🎯 Exam Tip: Breaking down complex logical transformations into smaller, sequential steps (like converse first, then inverse) helps prevent errors. Always write out the intermediate step.

 

Question 8. Write the converse of the inverse of p \( \implies \) q.
Answer:
Given statement: \( p \implies q \)
First, find its inverse: \( \sim p \implies \sim q \)
Next, find the converse of the inverse \( (\sim p \implies \sim q) \). To do this, we swap the two parts of the inverse.
Converse of inverse: \( \sim q \implies \sim p \). This is the same as the contrapositive of the original statement.
In simple words: First, add "not" to both parts of the original statement. Then, swap these new "not" parts.

🎯 Exam Tip: Notice that the "converse of the inverse" and the "inverse of the converse" both result in the contrapositive of the original statement. This is a good property to remember.

 

Question 9. Write the contrapositive of the inverse of p \( \implies \) q.
Answer:
Given statement: \( p \implies q \)
First, find its inverse: \( \sim p \implies \sim q \)
Next, find the contrapositive of the inverse \( (\sim p \implies \sim q) \). To do this, we negate both parts of the inverse and swap them.
Contrapositive of inverse: \( \sim (\sim q) \implies \sim (\sim p) \)
This simplifies to: \( q \implies p \). This is the converse of the original statement.
In simple words: First, make the inverse by adding "not" to both parts. Then, take this inverse, swap its parts, and add "not" to each of them again. Two "nots" cancel each other out.

🎯 Exam Tip: Double negations `~(~X)` always simplify to `X`. Keep an eye out for these to simplify your final logical expressions.

 

Question 10. Write the converse of the contrapositive of p \( \implies \) q.
Answer:
Given statement: \( p \implies q \)
First, find its contrapositive: \( \sim q \implies \sim p \)
Next, find the converse of the contrapositive \( (\sim q \implies \sim p) \). To do this, we swap the two parts of the contrapositive.
Converse of contrapositive: \( \sim p \implies \sim q \). This is the inverse of the original statement.
In simple words: First, find the contrapositive by swapping the parts and adding "not" to both. Then, take this new statement and just swap its "if" and "then" parts.

🎯 Exam Tip: Understanding that the contrapositive of \( p \implies q \) is \( \sim q \implies \sim p \) is fundamental. Then, applying the converse operation (swapping) to this derived statement is straightforward.

 

Question 11. Write the contrapositive of the contrapositive of p \( \implies \) q.
Answer:
Given statement: \( p \implies q \)
First, find its contrapositive: \( \sim q \implies \sim p \)
Next, find the contrapositive of this first contrapositive \( (\sim q \implies \sim p) \). To do this, we negate both parts of \( \sim q \implies \sim p \) and swap them.
Contrapositive of the contrapositive: \( \sim (\sim p) \implies \sim (\sim q) \)
This simplifies to: \( p \implies q \). This means that doing the contrapositive operation twice brings you back to the original statement. It's like a logical "undo" button.
In simple words: If you take a statement, flip it and make both parts "not", and then do that exact same thing again to the new statement, you end up right back where you started with the original statement.

🎯 Exam Tip: The double application of the contrapositive operation returning the original statement is a key property of logical equivalence. It shows that \( (p \implies q) \equiv (\sim q \implies \sim p) \equiv (p \implies q) \).

 

Question 12. Does completing each of the problems 6 through 10 result in a conditional? What is the relationship of each resulting condition to the original conditional P \( \implies \) q?
Answer:
Yes, completing each of the problems 6 through 10 results in a conditional statement.
The relationships to the original conditional \( P \implies q \) are:
Problem 6: Converse (\( q \implies \sim p \)), Inverse (\( p \implies \sim q \)), and Contrapositive (\( \sim q \implies p \)) of \( \sim p \implies q \).
Problem 7: Inverse of the converse of \( p \implies q \) is the contrapositive (\( \sim q \implies \sim p \)).
Problem 8: Converse of the inverse of \( p \implies q \) is the contrapositive (\( \sim q \implies \sim p \)).
Problem 9: Contrapositive of the inverse of \( p \implies q \) is the converse (\( q \implies p \)).
Problem 10: Converse of the contrapositive of \( p \implies q \) is the inverse (\( \sim p \implies \sim q \)).
The initial condition is the original statement \( p \implies q \) itself.
In simple words: For questions 6 to 10, all the final answers are "if-then" statements. Each of these final statements is related to the first simple "if p, then q" statement in a special way, like being its inverse or its converse.

🎯 Exam Tip: This question tests your ability to sequentially apply logical transformations and identify the resulting logical form relative to the original statement. Practice these combinations to quickly recall their outcomes.

 

Question 13. If p and q are any two propositions then prepare the truth table for p \( \implies \) q, \( \sim q \implies \sim p \) and show that the above statements are equivalent. Hence, or otherwise determine which of the following two arguments is valid?
(i) Given : If you work hard, then you pass the course. Given: You did not work hard. Conclusion: You did not pass the course.
(ii) Given : If you work hard, then you pass the course. Given: You did not pass the course. Conclusion: You did not work hard.
Answer:
The truth table to show that \( p \implies q \) and \( \sim q \implies \sim p \) are equivalent is given below:

From column 3 (\( p \implies q \)) and column 6 (\( \sim q \implies \sim p \)), we can see that they have identical truth values. This confirms that \( p \implies q \) is equivalent to \( \sim q \implies \sim p \). This is the law of contraposition.

(i) Let p: You work hard. Let q: You pass the course.
Given 1: \( p \implies q \) (If you work hard, then you pass the course.)
Given 2: \( \sim p \) (You did not work hard.)
Conclusion: \( \sim q \) (You did not pass the course.)
This argument is in the form of the fallacy of "denying the antecedent." It is not a valid argument. Just because you didn't work hard doesn't automatically mean you failed, as other factors might be at play (e.g., you might be naturally brilliant or lucky).

(ii) Let p: You work hard. Let q: You pass the course.
Given 1: \( p \implies q \) (If you work hard, then you pass the course.)
Given 2: \( \sim q \) (You did not pass the course.)
Conclusion: \( \sim p \) (You did not work hard.)
This argument is in the form of "Modus Tollens," which is a valid argument. If passing the course requires working hard, and you did not pass, then you must not have worked hard.
In simple words: The first part shows that if you swap an "if-then" statement and put "not" in both parts, it means the exact same thing. For the arguments: the first one is like saying "If it rains, the ground gets wet. It did not rain. So the ground is not wet." This is not always true (maybe someone watered it). The second argument is like saying "If it rains, the ground gets wet. The ground is not wet. So it did not rain." This is true.

🎯 Exam Tip: Remember two important valid argument forms: Modus Ponens (\( p \implies q, p \implies q \)) and Modus Tollens (\( p \implies q, \sim q \implies \sim p \)). Also, be aware of two common fallacies: Denying the Antecedent (\( p \implies q, \sim p \implies \sim q \)) and Affirming the Consequent (\( p \implies q, q \implies p \)).