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Subset

From AoPSWiki

We say a set $A$ is a subset of another set $B$ if every element of $A$ is also an element of $B$ , and we denote this by $A \sub B$ . The empty set is a subset of every set, and every set is a subset of itself. The notation $A \subseteq B$ emphasizes that $A$ may be equal to $B$ , while $A \subsetneq B$ says that $A$ is any subset of $B$ other than $B$ itself.

The following is a true statement:

$\emptyset \sub \{1, 2\} \sub \mathbb{N} \sub \mathbb{Z} \sub \mathbb{Q} \sub \mathbb{R} \sub \mathbb{C} \sub \mathbb{C}\, \cup\{\textrm{Groucho, Harpo, Chico}\} \supset \{1, 2, i, \textrm{Groucho}\}$

The set of all subsets of a given set $S$ is called the power set of $S$ and is denoted $\mathcal{P}(S)$ or $2^S$ .

Example Problems

Introductory

Counting the number of subsets in a set

Intermediate

1992 AIME Problem 2

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Category: Set theory

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