Question

Let X={a,b,c,1,2,3}, A={a,b,c}, B={a,2,3}, and C={1,2,3}. Which of the following pairs of subsets form a partition of X? Select all that apply.

(A) {A,B}
(B) {A,C}
(C) {B,C}
(D) {A,B,C}

Answer 1

The pair {A,C} forms a partition of X(option B).

Step-by-step explanation:

To determine which pairs of subsets form a partition of X, we need to understand the concept of a partition. A partition of a set is a collection of subsets that are non-empty, mutually exclusive, and their union is equal to the original set.

Let's analyze each pair of subsets:

(A) {A,B}: The subsets A and B have elements in common, so they are not mutually exclusive. Therefore, this pair does not form a partition of X.

(B) {A,C}: The subsets A and C do not have any elements in common, so they are mutually exclusive. Their union, {a,b,c,1,2,3}, equals X. Therefore, this pair forms a partition of X.

(C) {B,C}: The subsets B and C have elements in common, so they are not mutually exclusive. Therefore, this pair does not form a partition of X.

(D) {A,B,C}: The subsets A, B, and C have some elements in common, so they are not mutually exclusive. Therefore, this pair does not form a partition of X.

Based on the analysis, the correct pairs that form a partition of X are: (B) {A,C}.