University of California, San Diego: Edward Bender and S. Williamson's "Sets, Equivalence and Order: Sets and Functions"

Study Definition 1 on page SF-1. Typically, when an object is defined in mathematics, we next define when two of those objects are equal. Then we define operations on those objects. Now, for the objects are sets. When are two sets equal?

If A and B are sets, and if a ∈ A, implies a ∈ B, then we say A is a subset of B, denoted A \subset  B. The number of subsets of a set A, is denoted 2A (the reason for this notation will become clear when you study functions.)

A = B, if and only if, A \subset  B and B \subset  A. A and B are assumed to be subsets of a universal set E. Ø is the empty set or the set that has no members.

Note that the order of the elements in a set does not change the set. Work sufficient examples to ensure that you completely understand the concept of set equality. 
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