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Set Theory: Basic Language

  1. Sep 29, 2010 #1
    I'm reading the book Basic Set Theory by: Azriel Levy as I thought it might help me better understand Group Theory and Matrix Math. I have read the first chapter a number of times but I keep getting hung up on some of the syntax of the basic language or language of first-order predicate calculus with equality.

    In the first two pages I get stuck on this:

    i) [tex] \exists x\left ( x \in y \land \phi \right ) [/tex]

    where this is read "there is an x in y such that phi!"

    Such that phi what? I mean there are a lot of things I understand about what is being laid out in the language of first order logic such as:

    ii) [tex] \phi \land \psi [/tex] is [tex] \lnot ( \lnot \phi \lor \psi ) [/tex]

    but again I have trouble with i).

    I really thought I would get somewhere with this book and still probably will over a lengthy period of time. Is there something else I should be addressing first? What does i) mean? Where are all these brackets coming from and where were they supposed to have been defined?

    iii) [tex] R[A]=\left \{ y| \left ( \exists x \in A \right ) \left ( <x, y> \in R \right ) \right \} [/tex]

    There were lengthier examples! I think the brackets and the sudden realization of functions of the basic language are my two biggest hangups in being able to understand the full depth of the axioms being presented; many of which I have some vague understanding of from their general use in other subjects. Advice [tex] \land \lor [/tex] explanation of my above two dilemmas? :/

    Last edited: Sep 29, 2010
  2. jcsd
  3. Sep 29, 2010 #2


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    "such that phi" means "such that phi is true". It might seem weird to neglect the "is true" part, but when you think about actual statements like this it makes sense.

    For example "There exists x in the real numbers such that x2=2". You wouldn't say There exists x in the real numbers such that x2=2 is true" (here phi is the statement [tex]x^2=2[/tex])

    I don't know what the third part is supposed to be about, maybe some added context would be helpful? My first guess is that R is a relationship and <x,y> is just an ordered pair
  4. Sep 29, 2010 #3


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    The typical application of the product defined in (iii) is to talk about the image of a function or relation.

    If f is the function from the reals to itself defined pointwise by f(x) = x + 3, then we often encode f as its graph -- the set of all pairs (x,y) such that y = f(x).

    So [itex]f = \{ \langle x, x+3 \rangle \mid x \in \mathbb{R} \}[/itex]

    If A is the set {1} and B is the set [0,1] (the interval of real numbers), then can you tell me what f[A] and f are?
  5. Sep 30, 2010 #4
    With respect to iii) it was an example where I am becoming confused with respect to the usage of brackets and what they signify. Different brackets seem to have different meanings at different times.

    In this case the square brackets are addressing the class [tex] A [/tex] where the definition in the text for the use of the square brackets is to distinguish [tex] x [/tex] as being a subset or a () [tex] Dom(F) [/tex]. I understand that these <> brackets denote ordered pairs but [tex] < x | x \in V > [/tex] leaves me asking what the exact significance of the use of the brackets chosen is .... I'm sure I'll pick it up it's just been so hard to learn the things I want to learn because I don't have a proper grasp of the generally accepted language used to describe.

    Thanks for your replies and for the response to i). Both of your general explanations made things a little clearer for me. :)
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