Subsets

Every subset is a smaller set living within a larger one—like a slice of the whole pie! Understanding subsets helps us explore relationships and structure within sets. 🥧

Subsets

Suppose A and B are sets.

If every element of A is also an element of B, then we say A is a subset of B, denoted A \subseteq B.

We write A \not\subseteq B if A is not a subset of B, that is, if it is not true that every element of A is also an element of B.

Thus A \not\subseteq B means that there is at least one element of A that is not an element of B.

\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{R}

A = \{1,2\}, B = \{2,3,4\} A \not\subseteq B

Concept Check 1

Which of the following lists all of the subsets of the following set:

\{1,\{2,3\}\}

The correct answer is \{\}, \{1\}, \{\{2,3\}\}, \{1,\{2,3\}\}. Note that the empty set is always a subset of any set, and that the set \{2,3\} is treated as a single element in this context.