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

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Recall that the notation \(B \subseteq C\) means that every element of set \(B\) is also an element of set \(C\).

List the elements of set \(B\): \(\{2, 4, 8, 10\}\).

List the elements of set \(C\): \(\{4, 10, 12\}\).

Check each element of \(B\) to see if it is in \(C\): verify if \(2\) is in \(C\), if \(4\) is in \(C\), if \(8\) is in \(C\), and if \(10\) is in \(C\).

If all elements of \(B\) are found in \(C\), then \(B \subseteq C\) is true; if any element of \(B\) is not in \(C\), then \(B \subseteq C\) is false.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Subset Definition

A set B is a subset of set C (denoted B ⊆ C) if every element of B is also an element of C. This means no element in B can be outside of C. Checking subset relations involves verifying membership of all elements of B in C.

Set Membership

Set membership refers to whether a particular element belongs to a set. To determine if B ⊆ C, each element of B must be checked against the elements of C to confirm inclusion. This is fundamental in comparing sets.

Universal Set Context

The universal set U contains all elements under consideration. While U is not directly involved in subset checking here, understanding U helps clarify the scope of elements and ensures no elements outside U are mistakenly considered.

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