Let A = {2, 4, 6, 8, 10, 12}, B = {2, 4, 8, 10}, C = {4, 10, 12}, D = {2, 10}, andU = {2, 4, 6, 8, 10, 12, 14}. Determine whether each statement is true or false. B ⊆ C

Verified step by step guidance

1. Identify the elements of set B and set C. Set B = {2, 4, 8, 10} and set C = {4, 10, 12}.
2. Understand the definition of a subset. A set X is a subset of set Y if every element of X is also an element of Y.
3. Compare each element of set B to the elements of set C to check if all elements of B are present in C.
4. Notice that while 4 and 10 are in both B and C, the elements 2 and 8 are in B but not in C.
5. Conclude that since not all elements of B are in C, B is not a subset of C.

Key Concepts

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. In this context, understanding the relationships between sets, such as subsets, unions, and intersections, is crucial. A subset is defined as a set where all its elements are also contained in another set. This foundational concept is essential for evaluating statements about the relationships between sets A, B, C, and D.

Subset Notation

Subset notation is used to express the relationship between two sets, indicating that all elements of one set are contained within another. The notation 'B ⊆ C' means that set B is a subset of set C. To determine the truth of this statement, one must check if every element in set B is also present in set C, which is a key step in solving the problem.

Element Membership

Element membership refers to whether a specific element belongs to a set. In this question, it is important to analyze the elements of sets B and C to ascertain if all elements of B are included in C. This concept is fundamental in set theory and is used to verify subset relationships, which is necessary for determining the truth of the statement B ⊆ C.