Sets are unordered collections of related members in
which no members occur more than once. Formally, sets are written with
braces around them. Thus, if *S* is a set
containing the members 1, 2, and 3, then *S* = {1,
2, 3}. Of course, because a set is unordered, this is the same as
writing *S* = {3, 2, 1}. If a member,
*m*, is in a set, *S*, then membership is indicated
by writing *m* ∈ *S* ; otherwise,
*m* ∉
*S*. For example, in the set *S*
= {1, 2, 3}, 2 ∈
*S*, but 4 ∉ *S*. To effectively use
sets, we should be familiar with some definitions, basic operations,
and properties.

A set containing no members is the

*empty set*. The set of all possible members is the*universe*. (Of course, sometimes the universe is difficult to determine!) In set notation:Two sets are

*equal*if they contain exactly the same members. For example, if*S*_{1}= {1, 2, 3},*S*_{2}= {3, 2, 1}, and*S*_{3}= {1, 2, 4}, then*S*_{1}is equal to*S*_{2}, but*S*_{1}is not equal to*S*_{3}. In set notation:One set,

*S*_{1}, is a*subset*of another set,*S*_{2}, if*S*_{2}contains all of the members of*S*_{1}. For example, if*S*_{1}= {1, 3},*S*_{2}= {1, 2, 3}, and*S*_{3}= {1, 2}, then*S*_{1}is a subset of*S*_{2}, but*S*_{1}is not a subset of*S*_{3}. In set notation,

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