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Questions › Mathematics › Logic and proof techniques

Logic and proof techniques: previous year questions

Asked 73 times in 3 exams across 14 years. Newest first.

Practise Logic and proof techniques, easy to hard →

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  • Propositional logic and conditionals 15
  • Necessary and sufficient conditions 2
  • Quantifiers and their negation 6
  • Proof techniques 3
  • Logical reasoning puzzles 47

Propositional logic and conditionals

Statements, truth values and how if-then statements are negated, reversed and chained.

  1. JNU 2015 Q58Let X, Y, Z be statements. Suppose we know that ‘if X then Y ’ is true, and that ‘if Y then Z ’ is true. We also know that Y is false. We can infer…
  2. JNU 2015 Q59Let X and Y be statements. If we want to disprove the claim that ‘ X implies Y ’, we need to show that
  3. JNU 2015 Q60Let X, Y, Z be statements. Suppose we know that ‘ X implies Y ’, and that ‘ Z implies X ’. We also know that Y is false. We can infer that
  4. JNU 2015 Q62Let X and Y be statements. If we know that ‘ X implies Y ’, then we can also conclude that
  5. JNU 2015 Q63Let X, Y, Z be statements. Suppose we know that ‘ X implies Y ’, and that ‘ Y implies Z ’. If we also know that X is false, we can infer that
  6. JNU 2014 Q32Which of the following is true?
  7. JNU 2013 Q23"All those who have a good voice are good singers." This statement logically implies
  8. JNU 2013 Q83A theorem states that ‘if P then Q ’. From empirical observations it is known that P is false. Therefore it can be inferred that
  9. JNU 2012 Q40Three trucks A, B and C are used for transporting wheat and rice. The following information is given: (i) If A carried wheat, then B carried rice.…
  10. JNU 2011 Q43Let X, Y, Z be statements. Suppose we know that ' X is true implies Y is true', and ' X is false implies Z is true'. If we know that Z is false…
  11. JNU 2011 Q44Let X, Y, Z be statements. Suppose we know that X implies Y, and that Y implies Z. If we also know that Y is false, we can conclude that
  12. JNU 2010 Q67Let X and Y be statements. If we want to disprove the claim that " X implies Y ", we need to show that
  13. JNU 2010 Q68Let X, Y and Z be statements. Suppose we know that X implies Y, and that Z implies X. We also know that Y is false. We can infer that
  14. JNU 2010 Q71Let X and Y be statements. If we know that X implies Y, then we can also conclude that
  15. JNU 2010 Q72Let X, Y and Z be statements. Suppose we know that X implies Y, and that Y implies Z. If we also know that X is false, we can infer that

Necessary and sufficient conditions

Telling apart what must hold from what is enough to guarantee a result.

  1. CUET-PG 2023 Q100Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R. Assertion A: If a proposition P implies…
  2. JNU 2013 Q61If A is sufficient for B to occur and C is necessary for B to occur

Quantifiers and their negation

Statements with 'for all' and 'there exists', and how they change when negated or reordered.

  1. JNU 2011 Q42Let P(n, m) be a property about two integers n and m. If we want to disprove the claim that 'for every integer n, there exists an integer m such…
  2. JNU 2010 Q70Let P(n, m) be a property about two integers n and m. If we want to prove that "for every integer n, there exists an integer m such that P(n, m) is…
  3. DSE 2007 Q14If the statement 'There exists a legislature and a party such that every legislator in that party pays taxes' is false, then which of the following…
  4. DSE 2006 Q53Consider the following claim: 'There is some general election and some party such that all the candidates of that party in that election are…
  5. DSE 2005 Q13A number of mathematicians in the middle of the 20th century contributed to a series of books published in the name of a fictitious mathematician…
  6. DSE 2004 Q2Suppose an astronomer asserts 'Every solar system in the Milky Way galaxy contains a planet such that all the moons revolving around that planet…

Proof techniques

The standard ways to prove or disprove a claim.

  1. JNU 2018 Q47Let D = a² + b² + c², where a and b are successive positive integers and c = ab. Prove that √D is an odd positive integer.
  2. JNU 2015 Q61Let X and Y be statements. Which of the following strategies is not a valid way to show that ‘ X implies Y ’?
  3. JNU 2010 Q69Let X and Y be statements. Which of the following strategies is not a valid way to show that " X implies Y "?

Logical reasoning puzzles

Deduction problems that ask what must follow from a set of constraints.

  1. DSE 2021 Q14Three boxes are presented to a contestant. Exactly one box has an expensive gift and the other two are empty. Each box has a declaration imprinted…
  2. JNU 2018 Q20A student studying the weather for x days observed that (i) it rained on 7 days, morning or afternoon; (ii) when it rained in the afternoon, it was…
  3. JNU 2015 Q36Which of the following is a possible sequence of combinations for the salesman to visit in two successive years?
  4. JNU 2015 Q37If the salesman visits Pune, Bengaluru and Chandigarh in the first year, which of the following combinations must be visited in the third year?
  5. JNU 2015 Q38If the salesman visits Pune, Lucknow and Bhopal in the first year, which of the following combinations must be visited in the eleventh year?
  6. JNU 2014 Q33Assuming everybody’s testimony to be true, who is innocent and who is guilty?
  7. JNU 2014 Q34Answer Question Nos. 32–34 on the basis of the following information: Three individuals, A, B and C, are suspected of income tax evasion. They…
  8. JNU 2013 Q10Which of the following is a complete group of people who could be in?
  9. JNU 2013 Q11If B is out, then who must be in?
  10. JNU 2013 Q12If E and G are in the club, then what other two people must also be in the club?
  11. JNU 2013 Q13If B and D are out of the club, then which of the following must be true?
  12. JNU 2013 Q14If seven people are in the club, then who could be out?
  13. JNU 2013 Q15Who could be the only person in the club?
  14. JNU 2013 Q16Which of the follwoing species were the last two to go extinct?
  15. JNU 2013 Q17Which of the following species could have been the fourth one to go extinct?
  16. JNU 2013 Q18Which is the smallest number of species that could have gone extinct before T?
  17. JNU 2013 Q19Which of the following must be true if X went extinct after T?
  18. JNU 2013 Q20How many possible orders are there for the species’ extinction pattern?
  19. JNU 2013 Q21If the constraints were changed and X were not required to become extinct before U, then which of the following is a possible order for the species’…
  20. JNU 2013 Q46Which of the following is a possible sequence of combinations for the student in the two successive weeks?
  21. JNU 2013 Q47If the student studies Philosophy, Biology and Economics in the first week, which of the following combinations must be studied in the third week?
  22. JNU 2013 Q48If the student studies Philosophy, Literature and Mathematics in the first week, which of the following combinations must be studied in the eleventh…
  23. JNU 2013 Q84Following are given: (i) All P are Q. (ii) No Q is R. From (i) and (ii) we can infer that
  24. JNU 2013 Q85Following are given: (i) Some P are Q. (ii) No R is Q. From (i) and (ii) we can infer that
  25. JNU 2012 Q39X is a positive integer satisfying the following conditions: (i) 50 ≤ X ≤ 79 (ii) If X is a multiple of 3, then 50 ≤ X ≤ 59 (iii) If X is not a…
  26. JNU 2012 Q41Only one of the following three statements regarding the number of balls in an urn is true: (i) There are at least 100 balls in the urn. (ii) There…
  27. JNU 2012 Q42Q opted for
  28. JNU 2012 Q43Economics was opted by
  29. JNU 2012 Q44Mr. D is on the
  30. JNU 2012 Q45The person belonging to the cricket team is
  31. JNU 2011 Q45Suppose one wishes to prove that 'if some X are Y, then some Z are W '. To do this, it would suffice to show that
  32. JNU 2011 Q91The person who cheated in the examination was
  33. JNU 2011 Q92The accomplice of the person who cheated in the examination was
  34. JNU 2011 Q93There are four candidates for an award— A, B, C and D Only one of the four candidates had cleared both Tests I and II. Only one candidate had…
  35. JNU 2011 Q98On which day does S teach?
  36. JNU 2011 Q99Which subject does T teach?
  37. JNU 2011 Q100On which day is Economics taught?
  38. JNU 2010 Q46In case the painter works on Thursday, which among the following alternatives is possible?
  39. JNU 2010 Q47In case the painter works on Friday, which among the following statements must be untrue?
  40. JNU 2010 Q48Which arrangement among the following is possible?
  41. JNU 2010 Q62Which of the following is a possible sequence of combinations for the student in the two successive weeks?
  42. JNU 2010 Q63If the student studies Philosophy, Biology and Economics in the first week, which of the following combinations must be studied in the third week?
  43. JNU 2010 Q64If the student studies Philosophy, Literature and Mathematics in the first week, which of the following combinations must be studied in the eleventh…
  44. JNU 2010 Q66Suppose one wishes to prove that "if all X are Y, then all Z are W ". To do this, it would suffice to show that
  45. DSE 2009 Q22The WOW Language has only 2 letters, O and W, and obeys: (i) deleting successive letters WO from any word with more than 2 letters gives another…
  46. DSE 2009 Q30There are 3 persons A, B and C. One is a Truth-teller (always truthful), one a Liar (always lies), and the third a Normal person (sometimes lies…
  47. DSE 2007 Q50'Beautiful cities repel people. Look at India's most beautiful cities: they are among the least populated cities of the country.' Which of the…