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.