Question

Write each of the following sets by listing its elements explicitly.

a. [−4, 4] ∩ Z

b. (−4, 4] ∩ Z

c. (−4, [infinity]) ∩ Z

d. (−[infinity], 4] ∩ N

e. (−4, [infinity]) ∩ Z-

f. (4,5) ∩ Z

Answer 1

The question involves finding the set of integers (Z) and natural numbers (N) within certain intervals. For each part of the question, the elements of the intersection are listed, with attention to whether the interval is open or closed, and recognizing when the resulting set is empty or potentially infinite.

Step-by-step explanation:

The question concerns the intersection of sets with the set of integers Z and the set of natural numbers N. The notation used indicates intervals and the intersection symbol ∩ represents the numbers that are in both sets.

1. a. [−4, 4] ∩ Z represents all integers between -4 and 4, inclusive. Thus, the set is {-4, -3, -2, -1, 0, 1, 2, 3, 4}.
2. b. (−4, 4] ∩ Z represents all integers between -4 and 4, exclusive of -4 and inclusive of 4. Thus, the set is {-3, -2, -1, 0, 1, 2, 3, 4}.
3. c. (−4, ∞) ∩ Z represents all integers greater than -4. As this is an infinite set, we cannot list all elements, but it begins with {-3, -2, -1, 0, 1, 2, 3, ...}.
4. d. (−∞, 4] ∩ N represents all natural numbers up to and including 4. The set is {1, 2, 3, 4} as natural numbers start from 1.
5. e. (−4, ∞) ∩ Z is the same as c. and represents all integers greater than -4. The set begins with {-3, -2, -1, 0, 1, 2, 3, ...}.
6. f. (4,5) ∩ Z represents all integers strictly between 4 and 5. Since there are no integers between 4 and 5, the set is empty, so {} or ∅.