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Quiz-8

Page history last edited by Olivier P. Gerard 12 years, 5 months ago Saved with comment

Ce test est conçu pour tester vos connaissances en logique mathématique élémentaire (connecteurs logiques comme la négation, l'implication, ainsi que les quantificateurs).

 

(Key: correct, incorrect, partially correct.)

 

  1. Soient X et Y des affirmations. Si nous savons que X implique Y, alors nous pouvons en conclure que
    1. X est vrai et Y aussi.
    2. Y ne peut pas être faux. 
    3. Si Y est vrai, alors X est vrai. 
    4. Si Y est faux, alors X est faux.
      • CORRECT.
    5. Si X est faux, alors Y est faux.
    6. X ne peut pas être faux.
    7. Au moins une des deux affirmations (X, Y) est vraie. 
  2. Soient X et Y des affirmations. Si nous voulons démontrer comme faux l'énoncé "X et Y sont toutes les deux vraies", il nous faut montrer que
    1. Au moins l'une des deux affirmations (X, Y) est fausse. 
      • CORRECT.
    2. X et Y sont fausses toutes les deux. 
    3. X est fausse.
      • PARTIALLY. This will indeed disprove "Both X and Y are true", but X does not need to be false in order to disprove the above statement.
    4. Y is false.
      1. PARTIALLY. This will indeed disprove "Both X and Y are true", but Y does not need to be false in order to disprove the above statement.
      2. X does not imply Y, and Y does not imply X.
      3. Exactly one of X and Y are false.
      4. X is true if and only if Y is false.
  3. Let X and Y be statements. If we want to disprove the claim that "At least one of X and Y are true", we need to show that
    1. At least one of X and Y are false.
    2. X and Y are both false.
      • CORRECT.
    3. X is false.
    4. Y is false.
    5. X does not imply Y, and Y does not imply X.
    6. Exactly one of X and Y are false.
    7. X is true if and only if Y is false.
  4. Let X and Y be statements. If we want to disprove the claim that "X implies Y", we need to show that
    1. Y is true, but X is false.
    2. X is true, but Y is false.
      • CORRECT.
    3. X is false.
    4. Y is false.
    5. X and Y are both false.
    6. Exactly one of X and Y are false.
    7. At least one of X and Y is false.
  5. Let P(x) be a property about some object x of type X. If we want to disprove the claim that "P(x) is true for all x of type X", then we have to
    1. Show that there exists an x of type X for which P(x) is false.
      • CORRECT.
    2. Show that there exists an x which is not of type X, but for which P(x) is still true.
    3. Show that for every x in X, P(x) is false.
    4. Show that P(x) being true does not necessarily imply that x is of type X.
    5. Assume there exists an x of type X for which P(x) is true, and derive a contradiction.
    6. Show that there are no objects x of type X.
      • INCORRECT. Actually, if there are no objects of type X, then the statement "P(x) is true for all x of type X" is automatically true (but vacuously so)!
    7. Show that for every x in X, there is a y not equal to x for which P(y) is true.
  6. Let P(x) be a property about some object x of type X. If we want to disprove the claim that "P(x) is true for some x of type X", then we have to
    1. Show that there exists an x of type X for which P(x) is false.
    2. Show that there exists an x which is not of type X, but for which P(x) is still true.
    3. Show that for every x in X, P(x) is false.
      • CORRECT.
    4. Show that P(x) being true does not necessarily imply that x is of type X.
    5. Assume that P(x) is true for every x in X, and derive a contradiction.
    6. Show that there are no objects x of type X.
      • PARTIAL. This will certainly disprove the claim, however, one does not always need X to be empty in order to disprove the claim.
    7. Show that for every x in X, there is a y not equal to x for which P(y) is true.
  7. Let 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 true", then we should do the following:
    1. Let n be an arbitrary integer. Then find an integer m (possibly depending on n) such that P(n,m) is true.
      • CORRECT.
    2. Let n and m be arbitrary integers. Then show that P(n,m) is true.
      • INCORRECT. This will definitely prove what we want, but is far too strong, it proves much more than what we need!
    3. Find an integer n and an integer m such that P(n,m) is true.
    4. Let m be an arbitrary integer. Then find an integer n (possibly depending on m) such that P(n,m) is true.
    5. Find an integer n such that P(n,m) is true for every integer m.
    6. Find an integer m such that P(n,m) is true for every integer n.
      • INCORRECT. This will prove what we want, it is too strong - it proves more than we need.
    7. Show that whenever P(n,m) is true, then n and m are integers.
  8. Let 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 that P(n,m) is true", then we need to prove that
    1. There exists an integer n such that P(n,m) is false for all integers m.
      • CORRECT.
    2. There exists integers n,m such that P(n,m) is false.
    3. For every integer n, and every integer m, the property P(n,m) is false.
    4. For every integer n, there exists an integer m such that P(n,m) is false.
    5. For every integer m, there exists an integer n such that P(n,m) is false.
    6. There exists an integer m such that P(n,m) is false for all integers n.
    7. If P(n,m) is true, then n and m are not integers.
  9. Let P(n,m) be a property about two integers n and m. If we want to disprove the claim that "There exists an integer n such that P(n,m) is true for all integers m", then we need to prove that
    1. There exists an integer n such that P(n,m) is false for all integers m.
    2. There exists integers n,m such that P(n,m) is false.
    3. For every integer n, and every integer m, the property P(n,m) is false.
    4. For every integer n, there exists an integer m such that P(n,m) is false.
      • CORRECT.
    5. For every integer m, there exists an integer n such that P(n,m) is false.
    6. There exists an integer m such that P(n,m) is false for all integers n.
    7. If P(n,m) is true, then n and m are not integers.
  10. Let X and Y be statements. Which of the following strategies is not a valid way to show that "X implies Y"?
    1. Assume that X is true, and then use this to show that Y is true.
    2. Assume that Y is false, and then use this to show that X is false.
    3. Show that either X is false, or Y is true, or both.
    4. Assume that X is true, and Y is false, and deduce a contradiction.
    5. Assume that X is false, and Y is true, and deduce a contradiction.
      • CORRECT.
    6. Show that X implies some intermediate statement Z, and then show that Z implies Y.
    7. Show that some intermediate statement Z implies Y, and then show that X implies Z.
  11. Suppose one wishes to prove that "if all X are Y, then all Z are W". To do this, it would suffice to show that
    1. All Z are X, and all Y are W.
      • CORRECT.
    2. All X are Z, and all Y are W.
    3. All Z are X, and all W are Y.
    4. All X are Z, and all W are Y.
    5. All Y are X, and all W are Z.
    6. All Z are Y, and all X are W.
    7. All Y are Z, and all W are X.
  12. Suppose one wishes to prove that "if some X are Y, then some Z are W". To do this, it would suffice to show that
    1. All X are Z, and all Y are W.
      • CORRECT.
    2. Some X are Z, and all Y are W.
    3. All Z are X, and all Y are W.
    4. All X are Z, and some Y are W.
    5. Some Z are X, and some Y are W.
    6. Some Z are X, and all Y are W.
    7. All Z are X, and all W are Y.
  13. Let 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
    1. X is false.
    2. Z is false.
    3. X implies Z.
    4. B and C.
    5. A and C.
      • CORRECT.
    6. A, B, and C.
    7. None of the above conclusions can be drawn.
  14. Let X,Y,Z be statements. Suppose we know that X implies Y, and that Z implies X. If we also know that Y is false, we can conclude that
    1. X is false.
    2. Z is false.
    3. Z implies Y.
    4. B and C.
    5. A and C.
    6. A, B, and C.
      • CORRECT.
    7. None of the above conclusions can be drawn.
  15. Let 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, then we can conclude that
    1. X is false.
    2. X is true.
    3. Y is true.
    4. B and C.
      • CORRECT.
    5. A and C.
    6. A, B, and C.
    7. None of the above conclusions can be drawn.
    8.  
  16. Soient X,Y et Z des affirmations. On sait que X implique Y et que Y implique Z. Si d'autre part on sait que la proposition X est fausse, que peut-on en conclure ?
    1. Y est fausse.
    2. Z est fausse.
    3. Z implique X.
    4. A et B.
    5. A et C.
    6. A, B, et C.
    7. On ne peut tirer aucune conclusion.
      • CORRECT.

 

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