Insert ⊆ or ⊈ in each blank to make the resulting statement true. {1, 5} ____ {0, 1, 2, 3, 5}

Verified step by step guidance

1. Understand the symbols: The symbol \( \subseteq \) means 'is a subset of,' which means every element of the first set is also in the second set. The symbol \( s \) is not a standard set notation, so we focus on \( \subseteq \).
2. Identify the sets: The first set is \( \{1, 5\} \) and the second set is \( \{0, 1, 2, 3, 5\} \).
3. Check if every element of the first set is in the second set: Verify if 1 is in \( \{0, 1, 2, 3, 5\} \) and if 5 is in \( \{0, 1, 2, 3, 5\} \).
4. Since both 1 and 5 are elements of the second set, conclude that \( \{1, 5\} \subseteq \{0, 1, 2, 3, 5\} \) is a true statement.
5. Therefore, the correct symbol to insert in the blank is \( \subseteq \).

Key Concepts

Subset and Proper Subset

A subset is a set where every element of the first set is also in the second set, denoted by '⊆'. A proper subset means the first set is contained within the second set but is not equal to it, denoted by '⊂' or sometimes 's'. Understanding these symbols helps determine the correct relationship between two sets.

Set Elements and Membership

Set membership refers to whether an element belongs to a set. To determine subset relationships, you must check if all elements of the first set appear in the second set. For example, {1, 5} is a subset of {0, 1, 2, 3, 5} because both 1 and 5 are in the second set.

Equality of Sets

Two sets are equal if they contain exactly the same elements. If the first set has fewer elements than the second, it cannot be equal, only a proper subset. This concept helps decide whether to use '⊆' (subset or equal) or 's' (proper subset) in the statement.