Comment. Logic quiz
Comment. This quiz is designed to test your knowledge of the basics of mathematical logic (logical connectives such as negation and implication, as well as quantifiers).
Shuffle Questions.
Shuffle Answers.
Question. Let X and Y be statements. If we know that X implies Y, then we can also conclude that
Answer. X is true, and Y is also true.
Answer. Y cannot be false.
Answer. If Y is true, then X is true.
Correct Answer. If Y is false, then X is false.
Answer. If X is false, then Y is false.
Answer. X cannot be false.
Answer. At least one of X and Y is true.
Question. Let X and Y be statements. If we want to DISPROVE the claim that "Both X and Y are true", we need to show that
Correct Answer. At least one of X and Y are false.
Answer. X and Y are both false.
Partially Correct Answer. X is false.
Comment. 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.
Partially Correct Answer. Y is false.
Comment. 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.
Answer. X does not imply Y, and Y does not imply X.
Answer. Exactly one of X and Y are false.
Answer. X is true if and only if Y is false.
Question. 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
Answer. At least one of X and Y are false.
Correct Answer. X and Y are both false.
Answer. X is false.
Answer. Y is false.
Answer. X does not imply Y, and Y does not imply X.
Answer. Exactly one of X and Y are false.
Answer. X is true if and only if Y is false.
Question. Let X and Y be statements. If we want to DISPROVE the claim that "X implies Y", we need to show that
Answer. Y is true, but X is false.
Correct Answer. X is true, but Y is false.
Answer. X is false.
Answer. Y is false.
Answer. X and Y are both false.
Answer. Exactly one of X and Y are false.
Answer. At least one of X and Y is false.
Question. 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
Correct Answer. Show that there exists an x of type X for which P(x) is false.
Answer. Show that there exists an x which is not of type X, but for which P(x) is still true.
Answer. Show that for every x in X, P(x) is false.
Answer. Show that P(x) being true does not necessarily imply that x is of type X.
Answer. Assume there exists an x of type X for which P(x) is true, and derive a contradiction.
Answer. Show that there are no objects x of type X.
Comment. 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)!
Answer. Show that for every x in X, there is a y not equal to x for which P(y) is true.
Question. 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
Answer. Show that there exists an x of type X for which P(x) is false.
Answer. Show that there exists an x which is not of type X, but for which P(x) is still true.
Correct Answer. Show that for every x in X, P(x) is false.
Answer. Show that P(x) being true does not necessarily imply that x is of type X.
Answer. Assume that P(x) is true for every x in X, and derive a contradiction.
Partially Correct Answer. Show that there are no objects x of type X.
Comment. This will certainly disprove the claim, however, one does not always need X to be empty in order to disprove the claim.
Answer. Show that for every x in X, there is a y not equal to x for which P(y) is true.
Question. 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:
Correct Answer. Let n be an arbitrary integer. Then find an integer m (possibly depending on n) such that P(n,m) is true.
Answer. Let n and m be arbitrary integers. Then show that P(n,m) is true.
Comment. This will definitely prove what we want, but is far too strong, it proves much more than what we need!
Answer. Find an integer n and an integer m such that P(n,m) is true.
Answer. Let m be an arbitrary integer. Then find an integer n (possibly depending on m) such that P(n,m) is true.
Answer. Find an integer n such that P(n,m) is true for every integer m.
Answer. Find an integer m such that P(n,m) is true for every integer n.
Comment. This will prove what we want, it is too strong - it proves more than we need.
Answer. Show that whenever P(n,m) is true, then n and m are integers.
Question. 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
Correct Answer. There exists an integer n such that P(n,m) is false for all integers m.
Answer. There exists integers n,m such that P(n,m) is false.
Answer. For every integer n, and every integer m, the property P(n,m) is false.
Answer. For every integer n, there exists an integer m such that P(n,m) is false.
Answer. For every integer m, there exists an integer n such that P(n,m) is false.
Answer. There exists an integer m such that P(n,m) is false for all integers n.
Answer. If P(n,m) is true, then n and m are not integers.
Question. 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
Answer. There exists an integer n such that P(n,m) is false for all integers m.
Answer. There exists integers n,m such that P(n,m) is false.
Answer. For every integer n, and every integer m, the property P(n,m) is false.
Correct Answer. For every integer n, there exists an integer m such that P(n,m) is false.
Answer. For every integer m, there exists an integer n such that P(n,m) is false.
Answer. There exists an integer m such that P(n,m) is false for all integers n.
Answer. If P(n,m) is true, then n and m are not integers.
Question. Let X and Y be statements. Which of the following strategies is NOT a valid way to show that "X implies Y"?
Answer. Assume that X is true, and then use this to show that Y is true.
Answer. Assume that Y is false, and then use this to show that X is false.
Answer. Show that either X is false, or Y is true, or both.
Answer. Assume that X is true, and Y is false, and deduce a contradiction.
Correct Answer. Assume that X is false, and Y is true, and deduce a contradiction.
Answer. Show that X implies some intermediate statement Z, and then show that Z implies Y.
Answer. Show that some intermediate statement Z implies Y, and then show that X implies Z.
Question. 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
Correct Answer. All Z are X, and all Y are W.
Answer. All X are Z, and all Y are W.
Answer. All Z are X, and all W are Y.
Answer. All X are Z, and all W are Y.
Answer. All Y are X, and all W are Z.
Answer. All Z are Y, and all X are W.
Answer. All Y are Z, and all W are X.
Question. 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
Correct Answer. All X are Z, and all Y are W.
Answer. Some X are Z, and all Y are W.
Answer. All Z are X, and all Y are W.
Answer. All X are Z, and some Y are W.
Answer. Some Z are X, and some Y are W.
Answer. Some Z are X, and all Y are W.
Answer. All Z are X, and all W are Y.
// Now for the unshuffled-answer questions
Question. 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
Don't Shuffle Answers.
Answer. X is false.
Answer. Z is false.
Answer. X implies Z.
Answer. B and C.
Correct Answer. A and C.
Answer. A, B, and C.
Answer. None of the above conclusions can be drawn.
Question. 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
Don't Shuffle Answers.
Answer. X is false.
Answer. Z is false.
Answer. Z implies Y.
Answer. B and C.
Answer. A and C.
Correct Answer. A, B, and C.
Answer. None of the above conclusions can be drawn.
Question. 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
Don't Shuffle Answers.
Answer. X is false.
Answer. X is true.
Answer. Y is true.
Correct Answer. B and C.
Answer. A and C.
Answer. A, B, and C.
Answer. None of the above conclusions can be drawn.
Question. Let 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 conclude that
Don't Shuffle Answers.
Answer. Y is false.
Answer. Z is false.
Answer. Z implies X.
Answer. A and B.
Answer. A and C.
Answer. A, B, and C.
Correct Answer. No conclusion can be drawn.