Question

For each of the following, determine if:

(a) A is a subset of B
(b) B is a subset of A
(c) A is a proper subset of B
(d) B is a proper subset of A
(i) A = {1, 3, 5, 6, 8} B = {1, 5, 7}
(ii) A = {a, b, c, d} B = {d, c, a, b}
(iii) A = {2, 4, 5} B = x is an integer
(iv) A = {1, 3, 7} B = x is an odd integer

Answer 1

Set A is a subset of B if all elements of A are in B. If A has fewer elements than B, A is a proper subset of B. Examples provided illustrate these concepts for different sets.

Step-by-step explanation:

Determining Subsets and Proper Subsets

To determine if one set is a subset of another, we check if all elements of the first set are also in the second set. To determine if it is a proper subset, we additionally check if the two sets are not identical, meaning the subset has fewer elements than the set it is compared to.

1. A = {1, 3, 5, 6, 8}, B = {1, 5, 7}: A is not a subset of B since A contains elements not in B (3, 6, 8). Likewise, B is not a subset of A since it contains 7, which isn't in A. Neither is a proper subset of the other.
2. A = {a, b, c, d}, B = {d, c, a, b}: A and B are the same set, just written in a different order. Thus, A is a subset of B and B is a subset of A, but neither is a proper subset of the other.
3. A = {2, 4, 5}, B = x : A is a subset of B as all elements of A are integers. A is also a proper subset of B since B contains all integers, and A only has some integers.
4. A = {1, 3, 7}, B = x is an odd integer: A is a subset of B because all elements of A are odd integers. A is a proper subset of B because B contains all odd integers while A only contains some of them.