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In our first lecture on sets and set theory, we introduced a bunch of new symbols and terminology. This guide focuses on two of those symbols: ∈ and ⊆. These symbols represent concepts that, while related, are diferent from one another and can take some practice to get used to.
List of set symbols of set theory and probability. Table of set theory symbols. Symbol. Symbol Name. Meaning / definition. Example. { } set. collection of elements.
Definition: Given a set S, the power set of S is the set of all subsets of S. The power set is denoted by P(S). Examples: • Assume an empty set • What is the power set of ? P( ) = { } • What is the cardinality of P( ) ? | P( ) | = 1. • Assume set {1} • P( {1} ) = { , {1} } • |P({1})| = 2
We often de ne sets by listing their elements, or pairing down sets we already have by describing subsets via formulas. The symbol fx; y; zg describes the set whose elements are precisely x; y; and z. The symbol fx : p(x)g describes the set of all x such that the sentence p(x) is true.
To directly prove a universally quantified statement, allow the reader to pick an x and show that it has [some-property]. To directly prove an existentially quantified statement, specifically say what to pick for x and show that it has [some-property]. See also the Guide to Proofs.
Introduction. Every mathematician needs a working knowledge of set theory. The purpose of this chapter is to provide some of the basic information. Some additional set theory will be discussed in Chapter VIII. Sets are a useful vocabulary in many areas of mathematics. They provide a language for stating interesting results.
Set Theory Basics.doc 1.4. Subsets A set A is a subset of a set B iff every element of A is also an element of B. Such a relation between sets is denoted by A ⊆ B. If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B. (Caution: sometimes ⊂ is used the way we are using ⊆.) Both signs can be negated using the slash ...
