Insert ∈ or ∉ in each blank to make the resulting statement true. . 0 ____ ∅

Verified step by step guidance

1. Understand the symbols involved: \$0$ represents the number zero, and \(\emptyset\) (the empty set) is the set containing no elements.
2. Recall the meaning of the symbols \(\in\) and \(\notin\): \(a \in A\) means 'element \(a\) is in set \(A\)', and \(a \notin A\) means 'element \(a\) is not in set \(A\)'.
3. Determine if \$0\( is an element of the empty set \(\emptyset\). Since \(\emptyset\) has no elements, it cannot contain \)0$ or any other element.
4. Conclude that \(0 \notin \emptyset\) is the true statement because \$0$ is not an element of the empty set.
5. Therefore, the correct symbol to insert in the blank is \(\notin\).

Key Concepts

Set Membership (Element of a Set)

Set membership refers to whether an object is an element of a given set, denoted by the symbol ∈. If an object belongs to a set, we write the object ∈ set; if not, we write the object ∉ set. Understanding this helps determine if a specific item is contained within a set.

Empty Set (∅)

The empty set, denoted by ∅, is the unique set containing no elements. Since it has no members, no object can be an element of ∅. Recognizing the properties of the empty set is essential when evaluating membership statements involving ∅.

Distinction Between Elements and Subsets

It's important to distinguish between an element belonging to a set (∈) and a set being a subset of another (⊆). For example, 0 is an element, while {0} is a set containing 0. This distinction clarifies statements about membership versus subset relations.