Question

which set is a subset of every set?
○ ∅
○ {1}
○ {∅}

Explanation:

Step1: Recall the definition of a subset

A set \( A \) is a subset of a set \( B \) (denoted \( A \subseteq B \)) if every element of \( A \) is also an element of \( B \).

Step2: Analyze the empty set (\( \emptyset \))

The empty set has no elements. By the definition of a subset, since there are no elements in \( \emptyset \) to check if they are in another set, the statement "every element of \( \emptyset \) is in \( B \)" is vacuously true for any set \( B \). So \( \emptyset \subseteq B \) for any set \( B \).

Step3: Analyze \( \{1\} \)

Consider a set that does not contain 1, say \( B = \{2\} \). The element 1 is in \( \{1\} \) but not in \( \{2\} \), so \( \{1\}
subseteq \{2\} \). Thus, \( \{1\} \) is not a subset of every set.

Step4: Analyze \( \{\emptyset\} \)

Consider a set that does not contain \( \emptyset \), say \( B = \{1\} \). The element \( \emptyset \) is in \( \{\emptyset\} \) but not in \( \{1\} \), so \( \{\emptyset\}
subseteq \{1\} \). Thus, \( \{\emptyset\} \) is not a subset of every set.

Answer:

\( \emptyset \) (the first option, represented as \( \boxed{\emptyset} \) or in the given options as the first circle with \( \emptyset \))