Comment. Infimum and Supremum quiz
Comment. This quiz is designed to test your knowledge of order, and specifically
Comment. on infimum and supremum. All sets in this quiz are subsets of the real line R.
Shuffle Questions.
Shuffle Answers.
Question. Let A and B be bounded non-empty sets. Which of the following statements would be equivalent to saying that inf(A) <= inf(B)?
Correct Answer. For every b in B and epsilon > 0 there exists an a in A such that a < b + epsilon.
Partially Correct Answer. For every b in B there exists an a in A such that a <= b.
Comment. This will imply that inf(A) <= inf(B) but is not necessary (can you think of a counterexample?).
Answer. For every a in A there exists a b in B such that a <= b.
Answer. There exists a in A and b in B such that a < b.
Answer. For every a in A and every b in B, we have a <= b.
Partially Correct Answer. There exists a in A such that a <= b for all b in B.
Comment. This will imply that inf(A) <= inf(B) but is not necessary (can you think of a counterexample?).
Answer. There exists b in B such that a < b for all a in A.
Question. Let A and B be bounded non-empty sets such that inf(A) <= sup(B). Which of the following statements must be true?
Partially Correct Answer. There exists a in A and b in B such that a <= b.
Comment. Close, but not quite. Can you think of a counterexample?
Correct Answer. For every epsilon there exists a in A and b in B such that a < b+epsilon.
Answer. For every a in A and every b in B, we have a <= b.
Answer. For every a in A there exists a b in B such that a <= b.
Answer. For every b in B there exists an a in A such that a <= b.
Answer. There exists b in B such that a <= b for all a in A.
Answer. There exists a in A such that a <= b for all b in B.
Question. Let A and B be non-empty sets. Which of the following statements would be equivalent to saying that sup(A) <= inf(B)?
Answer. There exists a in A and b in B such that a <= b.
Correct Answer. For every a in A and every b in B, we have a <= b.
Answer. For every a in A there exists a b in B such that a <= b.
Answer. For every b in B there exists an a in A such that a <= b.
Answer. There exists b in B such that a <= b for all a in A.
Answer. There exists a in A such that a <= b for all b in B.
Answer. For every epsilon there exists a in A and b in B such that a < b+epsilon.
Question. Let A and B be non-empty sets. Which of the following statements would be equivalent to saying that sup(A) <= sup(B)?
Correct Answer. For every a in A and epsilon > 0 there exists a b in B such that a < b + epsilon.
Answer. For every a in A and every b in B, we have a <= b.
Partially Correct Answer. For every a in A there exists a b in B such that a <= b.
Comment. This implies that sup(A) <= sup(B) but is not necessary (can you think of a counterexample)?
Answer. For every b in B there exists an a in A such that a <= b.
Answer. There exists b in B such that a <= b for all a in A.
Answer. There exists a in A such that a <= b for all b in B.
Answer. For every epsilon there exists a in A and b in B such that a < b+epsilon.
Question. Let A and B be non-empty sets. The best we can say about sup(A union B) is that
Correct Answer. It is the maximum of sup(A) and sup(B).
Answer. It is the minimum of sup(A) and sup(B).
Answer. It is strictly greater than sup(A) and strictly greater than sup(B).
Answer. It is greater than or equal to sup(A) and greater than or equal to sup(B).
Answer. It is less than or equal to sup(A) and less than or equal to sup(B).
Answer. It is strictly greater than at least one of sup(A) and sup(B).
Answer. It is equal to at least one of sup(A) and sup(B).
Question. Let A and B be non-empty sets. The best we can say about sup(A intersect B) is that
Correct Answer. It is less than or equal to sup(A), and less than or equal to sup(B).
Answer. It is the maximum of sup(A) and sup(B).
Answer. It is the minimum of sup(A) and sup(B).
Answer. It is equal to at least one of sup(A) and sup(B).
Answer. It is strictly less than sup(A) and strictly less than sup(B).
Answer. It is greater than or equal to sup(A) and greater than or equal to sup(B).
Answer. It is greater than or equal to at least one of sup(A) and sup(B).
Question. Let A be a non-empty set. If sup(A) = +infinity, this means that
Correct Answer. For every real number M, there exists an a in A such that a > M.
Answer. There exists a in A and a real number M such that a > M.
Answer. There exists an a in A such that a > M for every real number M.
Answer. There exists a real number M such that a > M for every a in A.
Answer. A is the empty set.
Answer. For every real number M and every a in A, we have a > M.
Answer. There exists an a in A such that a > M for every real number M.
Question. Let A be a non-empty set. If inf(A) = +infinity, this means that
Answer. For every real number M, there exists an a in A such that a > M.
Answer. There exists a in A and a real number M such that a > M.
Answer. There exists an a in A such that a > M for every real number M.
Answer. There exists a real number M such that a > M for every a in A.
Correct Answer. A is the empty set.
Correct Answer. For every real number M and every a in A, we have a > M.
Comment. This answer is technically correct, but there is a simpler way to state it.
Answer. There exists an a in A such that a > M for every real number M.
Question. Let A be a non-empty set. If A is not bounded, this means that
Correct Answer. For every real number M, there exists an a in A such that |a| > M.
Answer. For every positive number M, there exists an a in A such that a > M, and for every negative number -M, there exists a' in A such that a' < -M.
Answer. sup(A) = +infinity and inf(A) = -infinity.
Answer. sup(A) = -infinity and inf(A) = +infinity.
Answer. For every real number M, there exists an a in A such that a > M.
Answer. There exists an a in A such that |a| > M for every real number M.
Answer. There exists a real number M such that |a| > M for every a in A.
Question. Let A and B be bounded non-empty sets. Which of the following statements would be equivalent to saying that sup(A) = inf(B)?
Answer. For every a in A and every b in B, we have a <= b.
Correct Answer. For every a in A and every b in B, we have a <= b. Also, for every epsilon > 0, there exists a in A and b in B such that b-a < epsilon.
Answer. For every epsilon > 0, there exists a in A and b in B such that b-a < epsilon.
Answer. There exists a real number L such that a <= L <= b for all a in A and b in B.
Answer. There exists a in A and b in B such that a <= b. However, for any epsilon > 0, there does not exist a in A and b in B for which a+epsilon <= b.
Answer. For every a in A there exists a b in B such that a+epsilon < b.
Answer. For every a in A there exists a b in B such that a+epsilon < b. Also, for every b in B there exists an a in A such that b+epsilon < a.
Question. Let A be a set, and let L be a real number. If sup(A) = L, this means that
Correct Answer. a <= L for every a in A. Also, for every epsilon > 0, there exists an a in A such that L - epsilon < a <= L.
Answer. a <= L for every a in A.
Answer. There exists an epsilon > 0 such that every real number a between L - epsilon and L lies in A.
Answer. Every number less than L lies in A, and every number greater than L does not lie in A.
Answer. L lies in A, and L is larger than every other element of A.
Comment. This implies that sup(A) = L, but is not necessary.
Answer. There exists a sequence x_n of elements in A which converges to L.
Answer. There exists a sequence x_n of elements in A which are less than L, but converges to L.
Question. Let A be a set, and let L be a real number. If sup(A) <= L, this means that
Answer. a <= L for every a in A. Also, for every epsilon > 0, there exists an a in A such that L - epsilon < a <= L.
Correct Answer. a <= L for every a in A.
Answer. There exists an epsilon > 0 such that every element of A is less than L - epsilon.
Answer. For every epsilon > 0 and every a in A, we have a <= L - epsilon.
Answer. For every epsilon > 0 there exists an a in A such that a <= L - epsilon.
Answer. There exists an a in A such that a <= L - epsilon for every epsilon > 0.
Answer. Every number less than or equal to L lies in A.
Question. Let A be a set, and let L be a real number. If sup(A) < L, this means that
Answer. a < L for every a in A. Also, for every epsilon > 0, there exists an a in A such that L - epsilon < a < L.
Answer. a < L for every a in A.
Comment. Close, but this is not quite sufficient to imply sup(A) < L (can you think of a counterexample?)
Correct Answer. There exists an epsilon > 0 such that every element of A is less than L - epsilon.
Answer. For every epsilon > 0 and every a in A, we have a < L - epsilon.
Answer. For every epsilon > 0 there exists an a in A such that a < L - epsilon.
Answer. There exists an a in A such that a < L - epsilon for every epsilon > 0.
Answer. Every number less than or equal to L lies in A.