1.
A set is:
Correct Answer
B. Well-defined collection of objects
Explanation
A set is a well-defined collection of objects. This means that a set must have clear and unambiguous criteria for determining whether an object belongs to the set or not. It cannot be vague or subjective. The objects in a set are referred to as elements of the set. Therefore, a set is a well-defined collection of objects where each object is an element of the set.
2.
The symbol ∈ means:
Correct Answer
C. Is an element of
Explanation
The symbol "∈" is used in set theory to indicate that an element belongs to a set. It represents the concept of membership, indicating that the element is part of the set.
3.
For larger sets:
Correct Answer
A. The description method is used, and it is more convenient
Explanation
The correct answer is "the description method is used, and it is more convenient." This means that for larger sets, the description method is utilized because it is considered to be more convenient. This suggests that the description method is preferred and offers advantages over other methods when dealing with larger sets.
4.
Z = {. . . ,−2,−1, 0, 1, 2, . . .} is the set of integers.
Correct Answer
A. True
Explanation
The statement is true because the set Z includes all integers, which are whole numbers both positive and negative, as well as zero. It is represented by the ellipsis notation, which indicates that the set continues indefinitely in both the negative and positive directions. Therefore, the given statement accurately describes the set of integers.
5.
N = {0, 1, 2, 3, . . .} is the set of integers.
Correct Answer
B. False
Explanation
The statement is not entirely accurate. The set N = {0, 1, 2, 3, ...} is typically used to represent the set of non-negative integers or whole numbers. It includes zero and all positive integers. However, it does not include negative integers. To represent the set of all integers, you would use Z = {..., -3, -2, -1, 0, 1, 2, 3, ...} which includes both positive and negative integers, along with zero. So, N represents non-negative integers, while Z represents all integers.
6.
All elements of all the sets under discussion belong to some universal set or universe.
Correct Answer
A. True
Explanation
This statement is true because in set theory, all sets are considered to be subsets of a universal set. The universal set contains all the elements that are being discussed in the context of the sets. Therefore, all elements of all the sets under discussion do indeed belong to some universal set.
7.
The empty set is the set containing no elements.
Correct Answer
A. True
Explanation
The empty set, also known as the null set, is a set that does not contain any elements. It is denoted by the symbol Ø or {} and is considered a subset of every set. Therefore, the statement "The empty set is the set containing no elements" is true.
8.
Let A and B be sets. A equals B, written A=B, iff every element of A is also an element of B, and conversely, every element of B is also an element of A.
Correct Answer
A. True
Explanation
The statement is true because it accurately defines the concept of set equality. According to the definition, two sets A and B are equal if and only if every element in A is also in B, and vice versa. This means that if A equals B, then any element that belongs to A must also belong to B, and any element that belongs to B must also belong to A. Therefore, the correct answer is true.
9.
Theorem. There is only one empty set, i.e., only two empty sets are equal.
Correct Answer
A. True
Explanation
This statement is true because the empty set is a unique set with no elements. It is defined as a set that contains no elements, and by definition, there can only be one set that meets this criteria. Therefore, there is only one empty set and any other set that is empty is equal to it.
10.
Let A and B be sets. Then A=B iff A belongs to B.
Correct Answer
B. False
Explanation
The statement "A=B iff A belongs to B" is not entirely accurate. The symbol "iff" stands for "if and only if," which implies a bidirectional or equivalence relationship. In set theory, "A=B" means that sets A and B have the same elements, but it doesn't imply that A belongs to B or vice versa.
If A belongs to B, it would mean that A is an element of B, not that A is equal to B. Likewise, if A is equal to B, it means that A and B have the same elements, but it doesn't necessarily mean that A is an element of B. Therefore, the statement "A=B iff A belongs to B" is not true in general for sets.