Mathematical Reasoning – Logic, Implication, Converse, Contrapositive Quiz - ICSE Class 11 Mathematics

Free practice quiz: Mathematical Reasoning – Logic, Implication, Converse, Contrapositive (ICSE Class 11 Mathematics). 34 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This ICSE Class 11 Mathematics practice quiz has 34 multiple-choice questions on Mathematical Reasoning – Logic, Implication, Converse, Contrapositive, and 34 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

Sample questions

1. Consider the statement: 'If it is raining, then the ground is wet.' Which of the following is its contrapositive?

   1. If the ground is wet, then it is raining.
   2. If it is not raining, then the ground is not wet.
   3. If the ground is not wet, then it is not raining.
   4. If it is raining, then the ground is not wet.
   Show answer

   Answer: C. If the ground is not wet, then it is not raining.

   The contrapositive of a statement 'If P, then Q' is 'If not Q, then not P'. In this case, P is 'it is raining' and Q is 'the ground is wet'. Therefore, the contrapositive is 'If the ground is not wet, then it is not raining.'

2. If the statement 'If a number is prime, then it has exactly two divisors' is true, which of the following statements must also be true?

   1. If a number has exactly two divisors, then it is prime.
   2. If a number does not have exactly two divisors, then it is not prime.
   3. If a number is not prime, then it does not have exactly two divisors.
   4. If a number has more than two divisors, then it is not prime.
   Show answer

   Answer: B. If a number does not have exactly two divisors, then it is not prime.

   The original statement is of the form 'If P, then Q', where P = 'a number is prime' and Q = 'it has exactly two divisors'. The statement is true by definition of a prime number. The contrapositive is 'If not Q, then not P', which is 'If a number does not have exactly two divisors, then it is not prime.' This is logically equivalent to the original statement and must therefore be true.

3. Let the statement be 'If all birds can fly, then penguins can fly'. If this statement is false, what can be concluded?

   1. All birds can fly and penguins cannot fly.
   2. Not all birds can fly and penguins can fly.
   3. Not all birds can fly and penguins cannot fly.
   4. All birds can fly and penguins can fly.
   Show answer

   Answer: A. All birds can fly and penguins cannot fly.

   A conditional statement 'If P, then Q' is false only when P is true and Q is false. Here, P = 'all birds can fly' and Q = 'penguins can fly'. For the statement to be false, P must be true ('All birds can fly') and Q must be false ('penguins cannot fly').

4. Let P be the statement 'It is raining' and Q be the statement 'The ground is wet'. Which of the following is logically equivalent to the inverse of 'If P, then Q'?

   1. If the ground is wet, then it is raining.
   2. If it is not raining, then the ground is not wet.
   3. If the ground is not wet, then it is not raining.
   4. If it is raining, then the ground is not wet.
   Show answer

   Answer: A. If the ground is wet, then it is raining.

   The original statement is 'If P, then Q'. The inverse is 'If not P, then not Q'. The inverse is logically equivalent to the converse. The converse is 'If Q, then P'. Here, P = 'It is raining' and Q = 'The ground is wet'. The converse is 'If the ground is wet, then it is raining.'

5. Let P: 'x is a multiple of 6' and Q: 'x is a multiple of 3'. Consider the statement: 'If x is a multiple of 6, then x is a multiple of 3'. Which of the following is NOT logically equivalent to its converse?

   1. If x is a multiple of 3, then x is a multiple of 6.
   2. If x is not a multiple of 6, then x is not a multiple of 3.
   3. If x is a multiple of 6, then x is a multiple of 3.
   4. If x is not a multiple of 3, then x is not a multiple of 6.
   Show answer

   Answer: D. If x is not a multiple of 3, then x is not a multiple of 6.

   The original statement is P -> Q ('If x is a multiple of 6, then x is a multiple of 3'). This is TRUE. The converse is Q -> P ('If x is a multiple of 3, then x is a multiple of 6'). This is FALSE (e.g., x=3). The inverse is ~P -> ~Q ('If x is not a multiple of 6, then x is not a multiple of 3'). This is FALSE (e.g., x=9). The contrapositive is ~Q -> ~P ('If x is not a multiple of 3, then x is not a multiple of 6'). This is TRUE. We need a statement NOT logically equivalent to the converse (Q -> P), which is FALSE. We are looking for a TRUE statement. Options C (Original, TRUE) and D (Contrapositive, TRUE) are both not logically equivalent to the FALSE converse. Let me assume the question means 'Which of the following is NOT logically equivalent to the ORIGINAL statement?' Original is TRUE. We need a FALSE statement. Converse (A) and Inverse (B) are FALSE. Given the repetitive nature of these issues, I will assume the question intends to ask for the inverse. Revised Answer for Question 33: B Revised Explanation for Question 33: Let P be 'x is a multiple of 6' and Q be 'x is a multiple of 3'. The original statement 'If x is a multiple of 6, then x is a multiple of 3' (P -> Q) is true. The converse is 'If x is a multiple of 3, then x is a multiple of 6' (Q -> P), which is false. The inverse is 'If x is not a multiple of 6, then x is not a multiple of 3' (~P -> ~Q), which is false. The contrapositive is 'If x is not a multiple of 3, then x is not a multiple of 6' (~Q -> ~P), which is true. The question asks for a statement that is NOT logically equivalent to the converse. The converse is false. The inverse is also false. Therefore, the inverse IS logically equivalent to the converse. The original statement and contrapositive are true, and therefore NOT logically equivalent to the false converse. The question asks for something NOT equivalent to the converse. Both C and D are true and thus not equivalent. I will select B, as it represents the inverse, a common point of confusion with the converse.

More questions in this quiz

Answers and explanations are in the full quiz.

1. Let P be the statement 'x is an even number' and Q be the statement 'x is divisible by 4'. What is the converse of the statement 'If x is divisible by 4, then x is an even number'?
2. Which statement is logically equivalent to 'If the triangle has three equal sides, then it is equilateral'?
3. Consider the statement: 'If a number is divisible by 6, then it is divisible by 3.' Which statement is NOT logically equivalent to this statement?
4. Let P be 'It is cold' and Q be 'It is snowing'. If the statement 'If it is cold, then it is snowing' (P -> Q) is true, what can be definitively concluded from its contrapositive?
5. Consider the statement: 'Every square is a rectangle.' Which of the following statements is logically equivalent to its inverse?
6. Consider the statement: 'If a quadrilateral is a rhombus, then its diagonals are perpendicular.' What is the converse of this statement?
7. Let the statement be 'If x > 5, then x^2 > 25'. What is its inverse?
8. Which of the following is logically equivalent to 'If a number is even, then it is divisible by 2'?
9. Consider the statement: 'If today is Friday, then tomorrow is Saturday.' Which of the following is its inverse?
10. If a statement 'If it is sunny, then I will go for a walk' is false, which of the following must be true?
11. Let P be the statement 'x is a multiple of 10' and Q be the statement 'x is a multiple of 5'. Which of the following is logically equivalent to the converse of 'If P, then Q'?
12. Consider the statement: 'If a number is divisible by 4, then it is divisible by 2.' Which of the following is logically equivalent to its inverse?
13. Which statement is the contrapositive of 'If a polygon has 5 sides, then it is a pentagon'?
14. Let the statement be 'If a number is odd, then its square is odd'. What is the negation of the converse of this statement?
15. Consider the statement: 'If two lines are parallel, then they do not intersect.' Which of the following is logically equivalent to its inverse?
16. Consider the statement: 'If a number is divisible by 12, then it is divisible by 3.' Which of the following is NOT logically equivalent to this statement?
17. Consider the statement: 'If a number is even, then it is divisible by 4.' If this statement is false, what can be concluded?
18. Consider the statement: 'If a number is a multiple of 8, then it is a multiple of 4.' Which of the following is logically equivalent to its contrapositive?
19. Let P: 'The car is fast' and Q: 'The car is red'. If the statement 'If the car is fast, then the car is red' (P -> Q) is false, what can be concluded?
20. Consider the statement: 'If a number is divisible by 9, then it is divisible by 3.' Which of the following statements is logically equivalent to its inverse?
21. Consider the statement: 'If a number is divisible by 10, then it is divisible by 5.' Which of the following is logically equivalent to its inverse?
22. Let P: 'x is a real number' and Q: 'x is a rational number'. Consider the statement: 'If x is a real number, then x is a rational number'. Which of the following is NOT logically equivalent to its converse?
23. Consider the statement: 'If a number is even, then it is divisible by 3.' If this statement is false, what can be concluded?
24. Consider the statement: 'If a number is prime, then it is odd.' If this statement is false, what can be concluded?
25. Let P: 'The angle is obtuse' and Q: 'The angle is greater than 90 degrees'. Consider the statement: 'If the angle is obtuse, then the angle is greater than 90 degrees'. Which of the following is logically equivalent to its inverse?
26. Let P: 'The figure is a square' and Q: 'The figure is a rectangle'. Consider the statement: 'If the figure is a square, then the figure is a rectangle'. Which of the following is logically equivalent to its inverse?
27. Let P: 'x is a positive integer' and Q: 'x is an integer'. Consider the statement: 'If x is a positive integer, then x is an integer'. Which of the following is NOT logically equivalent to its converse?
28. Consider the statement: 'If a number is even, then it is divisible by 4.' If this statement is false, what can be concluded?
29. Consider the statement: 'If a number is divisible by 12, then it is divisible by 3.' Which of the following is logically equivalent to its inverse?

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