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).
Discuss this quiz
(Key: correct, incorrect, partially correct.)
- Let X and Y be statements. If we know that X implies Y, then we can also conclude that
- X is true, and Y is also true.
- Y cannot be false.
- If Y is true, then X is true.
- If Y is false, then X is false.
- If X is false, then Y is false.
- X cannot be false.
- At least one of X and Y is true.
- 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
- At least one of X and Y are false.
- X and Y are both false.
- X is false.
- 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.
- Y is false.
- 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.
- X does not imply Y, and Y does not imply X.
- Exactly one of X and Y are false.
- X is true if and only if Y is false.
- 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
- At least one of X and Y are false.
- X and Y are both false.
- X is false.
- Y is false.
- X does not imply Y, and Y does not imply X.
- Exactly one of X and Y are false.
- X is true if and only if Y is false.
- Let X and Y be statements. If we want to disprove the claim that "X implies Y", we need to show that
- Y is true, but X is false.
- X is true, but Y is false.
- X is false.
- Y is false.
- X and Y are both false.
- Exactly one of X and Y are false.
- At least one of X and Y is false.
- 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
- Show that there exists an x of type X for which P(x) is false.
- Show that there exists an x which is not of type X, but for which P(x) is still true.
- Show that for every x of type X, P(x) is false.
- Show that P(x) being true does not necessarily imply that x is of type X.
- Assume there exists an x of type X for which P(x) is true, and derive a contradiction.
- 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)!
- Show that for every x of type X, there is a y not equal to x for which P(y) is true.
- 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
- Show that there exists an x of type X for which P(x) is false.
- Show that there exists an x which is not of type X, but for which P(x) is still true.
- Show that for every x of type X, P(x) is false.
- Show that P(x) being true does not necessarily imply that x is of type X.
- Assume that P(x) is true for every x of type X, and derive a contradiction.
- 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.
- Show that for every x of type X, there is a y not equal to x for which P(y) is true.
- 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:
- Let n be an arbitrary integer. Then find an integer m (possibly depending on n) such that P(n,m) is true.
- 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!
- Find an integer n and an integer m such that P(n,m) is true.
- Let m be an arbitrary integer. Then find an integer n (possibly depending on m) such that P(n,m) is true.
- Find an integer n such that P(n,m) is true for every integer m.
- Find an integer m such that P(n,m) is true for every integer n.
- INCORRECT. This will prove what we want, but it is too strong - it proves more than we need.
- Show that whenever P(n,m) is true, then n and m are integers.
- 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
- There exists an integer n such that P(n,m) is false for all integers m.
- There exists integers n,m such that P(n,m) is false.
- For every integer n, and every integer m, the property P(n,m) is false.
- For every integer n, there exists an integer m such that P(n,m) is false.
- For every integer m, there exists an integer n such that P(n,m) is false.
- There exists an integer m such that P(n,m) is false for all integers n.
- If P(n,m) is true, then n and m are not integers.
- 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
- There exists an integer n such that P(n,m) is false for all integers m.
- There exists integers n,m such that P(n,m) is false.
- For every integer n, and every integer m, the property P(n,m) is false.
- For every integer n, there exists an integer m such that P(n,m) is false.
- For every integer m, there exists an integer n such that P(n,m) is false.
- There exists an integer m such that P(n,m) is false for all integers n.
- If P(n,m) is true, then n and m are not integers.
- Let X and Y be statements. Which of the following strategies is not a valid way to show that "X implies Y"?
- Assume that X is true, and then use this to show that Y is true.
- Assume that Y is false, and then use this to show that X is false.
- Show that either X is false, or Y is true, or both.
- Assume that X is true, and Y is false, and deduce a contradiction.
- Assume that X is false, and Y is true, and deduce a contradiction.
- Show that X implies some intermediate statement Z, and then show that Z implies Y.
- Show that some intermediate statement Z implies Y, and then show that X implies Z.
- 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
- All Z are X, and all Y are W.
- All X are Z, and all Y are W.
- All Z are X, and all W are Y.
- All X are Z, and all W are Y.
- All Y are X, and all W are Z.
- All Z are Y, and all X are W.
- All Y are Z, and all W are X.
- 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
- All X are Z, and all Y are W.
- Some X are Z, and all Y are W.
- All Z are X, and all Y are W.
- All X are Z, and some Y are W.
- Some Z are X, and some Y are W.
- Some Z are X, and all Y are W.
- All Z are X, and all W are Y.
- 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
- X is false.
- Z is false.
- X implies Z.
- B and C.
- A and C.
- A, B, and C.
- None of the above conclusions can be drawn.
- 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
- X is false.
- Z is false.
- Z implies Y.
- B and C.
- A and C.
- A, B, and C.
- None of the above conclusions can be drawn.
- 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
- X is false.
- X is true.
- Y is true.
- B and C.
- A and C.
- A, B, and C.
- None of the above conclusions can be drawn.
- 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
- Y is false.
- Z is false.
- Z implies X.
- A and B.
- A and C.
- A, B, and C.
- No conclusion can be drawn.
Score:
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