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Basics of Set Theory

In this section we shall consider some elementary consepts related to sets

and their elements ,assuming that at a certain level ,the students have encountered the notions .In particular we wish to review (not necessarily in this order)

• 
  Element containment (E)
  
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  Containment relationships between sets (….. same as ),….(same as ))
  
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  Operations on subsets of a given set :intersection ( ,union ,( , difference (-),and symmetric difference (+) of two subsets of a given set.
  
• 
  Set-theoretic constructions: Power set (2 s),and Cartesian product (S x T)
  
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  Mappings (I,e,…,functions ) between sets.
  
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  Relations and equivalence relations on sets.
  

Looks scray ,doesn’t it? Don’t worry it’s all very natural…

Before we launch into these topics ,let’s get really crazy for a moment.What we’re going to talk about is naïve set theory .As oppesed to what, you might ask? Well, here’s the point.When talking about sets,we typically use the language,

“The set of all……”

Don’t we often talk like this? Haven’t you heard me say, ”consider the set of all integers” or “the set of all real numbers ”? Maybe I’ve even asked you to think about the “set of all differentiable functions defined on the whole real line”. Surely none of this can possibly cause any difficulties !But what if we decide to consider something really huge, like the “set of all sets”? Despite the fact that this set is really big, it shouldn’t be a problem,should it? The only immediately peculiar aspect of this set-let’s call it B (for “big”)-is that not only B… B (which is true for all set’s ), but also that B ⊂ B. Since the set {1} ⊂{1},we see that for a given set A , it may or may not happen that A ⊂ A .This leads us to consider, as did Bertrand Russell ,the set of all sets which don’t contain themselves as an element:in symbols we would write this as.

R={S | S ⊂ S},

This set R seems strange but is it really a problem? Well, let’s take a closer look ,asking the question ,is R є R? By looking at the definition, we see that R є R if any only if R ⊂ R! This is impossible! This is a paradox, often called Russell’s paradox (or Russsell’s Antinomy)

Conclusion: Naïve set theory leads to paradoxes! So what do we do? There are basically two choices: we could be much more careful and do axiomatic set theory ,a highly formalized approach to set theory (I don’t care for the theory, myself!) but one that is free of such paradoxes. A more sensible approach for us is simply to continue to engage in naïve set theory, trying to avoid sets that seem unreasonably large and hope for the best!