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. ∅ ⊆ ∅

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Recall the definition of a subset: A set \(X\) is a subset of a set \(Y\) (written as \(X \subseteq Y\)) if every element of \(X\) is also an element of \(Y\).

Consider the empty set \(\emptyset\). By definition, it has no elements.

Since there are no elements in \(\emptyset\), there are no elements that violate the condition of being in the other set when checking \(\emptyset \subseteq \emptyset\).

This means that the empty set is a subset of every set, including itself.

Therefore, the statement \(\emptyset \subseteq \emptyset\) is true.

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Key Concepts

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

Empty Set (∅)

The empty set, denoted by ∅, is the set containing no elements. It is unique and is a subset of every set, including itself. Understanding the empty set is fundamental in set theory and helps in evaluating subset relations.

Subset Definition (⊆)

A set A is a subset of set B (A ⊆ B) if every element of A is also an element of B. This includes the case where A is empty, since there are no elements to contradict the condition. Recognizing this helps determine the truth of subset statements.

Properties of the Empty Set in Subset Relations

The empty set is considered a subset of every set, including itself, because there are no elements in ∅ that violate the subset condition. This property is essential when evaluating statements like ∅ ⊆ ∅, which is always true.

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