Comment. Sets quiz
Comment. This quiz is designed to test your knowledge of basic concepts in sets.
Shuffle Questions.
Shuffle Answers.
Question. Let A and B be sets. What does it mean if we say that x is an element of (A union B)?
Answer. x is an element of A and is also an element of B.
Comment. This is what it means for x to lie in (A intersect B).
Answer. x is an element of A.
Comment. It is true that if x lies in A, then x also lies in (A union B); but it is possible to lie in (A union B) without lying in A.
Answer. x is an element of B.
Comment. It is true that if x lies in B, then x also lies in (A union B); but it is possible to lie in (A union B) without lying in B.
Correct Answer. x is an element of A, or is an element of B, or both.
Answer. A is equal to B, and x is an element of both.
Comment. One does not need the two sets A and B to be equal in order to form the union (A union B).
Answer. x is an element of A, or is an element of B, but not both.
Comment. If x lies in both A and B then it still qualifies to lie in (A union B).
Answer. x is equal to some element a of A plus some other element b of B: x = a + b.
Comment. This is what it means for x to lie in (A+B).
Question. Let A and B be sets. What does it mean if we say that x is NOT an element of (A union B)?
Correct Answer. x is not an element of A, and x is not an element of B.
Answer. Either x is not an element of A, or x is not an element of B.
Comment. This is what it means for x to not be an element of (A intersect B).
Answer. x does not belong to both A and B at the same time.
Comment. This is what it means for x to not be an element of (A intersect B).
Answer. There is some element of A which is not equal to x, and there is some element of B which is not equal to x.
Answer. There is some element of either A or B which is not equal to x.
Answer. A and B are the same sets, and x is not an element of either.
Question. Let A and B be sets. What does it mean if we say that x is an element of (A intersect B)?
Correct Answer. x is an element of A and x is also an element of B.
Answer. x is an element of A.
Answer. x is an element of B.
Answer. x is an element of A, or is an element of B, or both.
Comment. This is what it means for x to lie in (A union B).
Answer. A is equal to B, and x is an element of both.
Comment. One does not need the two sets A and B to be equal in order to form the intersection (A intersect B).
Answer. x is an element of A, or is an element of B, but not both.
Comment. This is what it means for x to lie in (A XOR B).
Answer. x is equal to some element a of A plus some other element b of B: x = a + b.
Comment. This is what it means for x to lie in (A+B).
Question. Let A and B be sets. What does it mean if we say that x is NOT an element of (A intersect B)?
Correct Answer. x cannot belong to both A and B; it may belong to A, or to B, or to neither, but not both.
Answer. x is not an element of A, and x is not an element of B.
Comment. This does not cover the possibility that x is an element of exactly one of A or B
Answer. Every element of A and every element of B is different from x.
Comment. This is what it means for x to not be an element of (A union B).
Answer. There is some element of A which is not equal to x, and there is some element of B which is not equal to x.
Answer. x belongs to exactly one of A and B.
Comment. This does not cover the possibility that x belongs to neither A nor B.
Answer. x belongs to neither A nor B.
Comment. This is what it means for x to not be an element of (A union B).
Answer. A and B are the same sets, and x is not an element of either.
Question. Let A and B be sets. What does it mean if we say that A is a subset of B?
Correct Answer. Every element x in A is also an element of B.
Answer. Every element y in B is also an element of A.
Comment. This is what it means for B to be a subset of A.
Answer. Every element x in A is equal to every element y in B.
Answer. Some element x in A is also an element of B.
Comment. This is what it means for A and B to have a non-empty intersection.
Answer. Every element x in A is contained in some element y of B.
Comment. We want the elements in A to be _equal_ to elements in B, not _contained_ in them.
Answer. Every element y of B is equal to some element x of A.
Comment. This is what it means for B to be a subset of A.
Answer. A is an element of B.
Comment. We want the _elements_ of A to be elements of B; we don't what A itself to be an element of B.
Question. Let A and B be sets. What does it mean if we say that A is NOT a subset of B?
Answer. B is a subset of A.
Comment. It is possible for A and B to not be subsets of each other.
Answer. B is equal to A.
Answer. B and A are disjoint.
Comment. It is possible for A and B to partially intersect without being subsets of each other.
Correct Answer. There is an element x of A which does not lie in B.
Answer. Every element x of A does not lie in B.
Comment. It is possible for A and B to still have common elements without A being a subset of B.
Answer. There is an element y of B which does not lie in A.
Comment. This is what it means for B to not be a subset of A.
Answer. A is not an element of B.
Question. Let A and B be sets. What does it mean if we say that A is equal to B?
Correct Answer. Every element x in A is also an element of B, and every element y in B is also an element of A.
Answer. Every element y in B is equal to some element of A.
Comment. This only shows that B is a subset of A.
Answer. Every element x in A is equal to every element y in B.
Comment. This only shows that A is a subset of B.
Answer. Some element x in A is equal to some element of B.
Comment. This only shows that A and B have some non-empty intersection.
Answer. A is not contained in B, and B is not contained in A.
Comment. If A and B are equal, then they are automatically contained in each other.
Answer. A is not strictly contained in B, and B is not strictly contained in A.
Comment. It is possible for A and B to be unequal, and to not be strictly contained in each other.
Answer. Every element in A is equal to every element in B.
Comment. This can only be true if A and B have at most one element.
Question. Let A and B be sets. What does it mean if we say that A and B are disjoint?
Correct Answer. There does not exist any element x which belongs to both A and B.
Answer. A is not a subset of B, and B is not a subset of A.
Answer. A is not equal to B.
Answer. The union of A and B is empty.
Answer. There exists an element x of A and an element y of B such that x is not equal to y.
Answer. There is an element x of A which is not in B, and there is an element y of B which is not in A.
Question. Let A and B be sets. What does it mean if we say that A and B are NOT disjoint?
Correct Answer. There exists an element x which belongs to both A and B.
Answer. Either A is a subset of B, or B is a subset of A.
Answer. A is a subset of B, and B is a subset of A.
Answer. A is equal to B.
Answer. Every element of A is equal to every element of B.
Answer. Every element of A is equal to some element of B, and vice versa.
Answer. The union of A and B is non-empty.
Question. Let A and B be sets. What does it mean if we say that A is NOT equal to B?
Correct Answer. Either there is some element x of A which is not in B, or there is some element y in B which is not in A, or both.
Answer. There is some element x of A which is not in B, and there is some element y in B which is not in A.
Answer. Either A is a subset of B, or B is a subset of A.
Answer. Either A is a proper subset of B, or B is a proper subset of A.
Answer. For every x in A and every y in B, x is not equal to y.
Answer. For every x in A there is some y in B such that x is not equal to y.
Answer. There is some x in A and some y in B such that x is not equal to y.
Question. Let A be the set { sin(x): 0 < x < pi }. What does it mean if we say that y is an element of A?
Comment. It turns out that A is in fact the half-open interval { y: 0 < y <= 1 } (why?). But you did not need to know that to work out this problem.
Correct Answer. y is equal to sin(x) for some 0 < x < pi.
Answer. y is equal to sin(x) for every 0 < x < pi.
Answer. sin(y) is between 0 and pi.
Answer. y is between 0 and pi.
Answer. y is between sin(0) and sin(pi).
Answer. sin(y) is an element of A.
Question. Let A be the set { sin(x): 0 < x < pi }. What does it mean if we say that y is NOT an element of A?
Answer. There exists 0 < x < pi such that y is not equal to sin(x).
Correct Answer. For every 0 < x < pi, y is not equal to sin(x).
Answer. sin(y) is not between 0 and pi.
Answer. y is not between 0 and pi.
Answer. y is not between sin(0) and sin(pi).
Answer. sin(y) is not an element of A.