Question

Sets A through C are defined as follows.

A = {1, 2, 3, 4}
B = {-1, -2, -3}
C = {-1, 0, 1, 2, 3}
Indicate whether each statement is true or false.

(a) |A ∩ B| = 1

(b) {1, 2} ⊂ P(A)
(c) A ∪ B = A ⊕ B

Answer 1

False for (a) as no common elements exist between A and B; true for (b) as {1, 2} is part of the power set of A; false for (c) as A ∪ B equals A ⊕ B when A and B have no common elements.

Step-by-step explanation:

The student is asking about the properties of set operations like intersection, union, and symmetric difference, using sets A, B, and C. To answer the question, let's evaluate each statement:

1. |A ∩ B| = 1: False. Sets A and B do not have any elements in common, so their intersection is empty, and the cardinality is 0, not 1.
2. {1, 2} ⊂ P(A): True. {1, 2} is a subset of set A, and thus it is an element of the power set of A, P(A), which contains all subsets of A.
3. A ∪ B = A ⊕ B: False. The union of A and B includes all elements from both sets, but the symmetric difference would only include elements that are in either A or B, but not in both. Since A and B have no common elements, A ⊕ B is the same as A ∪ B in this particular case.