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. {0, 2} ⊆ D

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

Identify the elements of the set \(\{0, 2\}\), which are \(0\) and \(2\).

Look at the elements of set \(D = \{2, 10\}\) and check if both \(0\) and \(2\) are in \(D\).

Since \(2\) is in \(D\) but \(0\) is not, not all elements of \(\{0, 2\}\) are in \(D\).

Therefore, conclude that \(\{0, 2\} \subseteq D\) is false because the element \(0\) is missing from \(D\).

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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 A is a subset of set B if every element of A is also an element of B. This means all members of the smaller set must be contained within the larger set for the subset relation to hold true.

Element Membership in Sets

Element membership refers to whether a specific element belongs to a set. To verify subset relations, each element of the potential subset must be checked for membership in the other set.

Empty Set and Zero Element

The empty set {} contains no elements, while 0 is a specific number that may or may not be in a set. Distinguishing between the empty set and sets containing zero is crucial when evaluating subset statements.

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