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Math Help - Group Theory - Describing Functions

  1. #1
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    Group Theory - Describing Functions

    Hi,

    I am working on the first exercise (I have 10) and I don't know what it is about... I would really appreciate it if you could give me some help on this one. Hints/solving a similar one...whatever, it'll help me solve the other exercises I have. Thanks a lot

    Exercise:

    Let X be a set (non necessary finite), let P(X) (power set of X) be the set of all subset of the set X, let Y be the set of all functions X → {0, 1}, let Φ: P(X) → Y be a function such that

    Φ(Σ)(x)=
    1 if x ∈ Σ,
    0 if x /∈ Σ (is not in Σ)

    for every subset Σ of the set X. Let us denote the empty set by the symbol ∅.

    describe the function Φ(X) and Φ(∅),
    describe Φ(Σ) in the case when X = Z and Σ is the set of all even integers,
    describe Φ(Σ) in the case when X = R and Σ = {0},
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mbmstudent View Post
    Hi,

    I am working on the first exercise (I have 10) and I don't know what it is about... I would really appreciate it if you could give me some help on this one. Hints/solving a similar one...whatever, it'll help me solve the other exercises I have. Thanks a lot

    Exercise:

    Let X be a set (non necessary finite), let P(X) (power set of X) be the set of all subset of the set X, let Y be the set of all functions X → {0, 1}, let Φ: P(X) → Y be a function such that

    Φ(Σ)(x)=
    1 if x ∈ Σ,
    0 if x /∈ Σ (is not in Σ)

    for every subset Σ of the set X. Let us denote the empty set by the symbol ∅.

    • describe the function Φ(X) and Φ(∅),
    • describe Φ(Σ) in the case when X = Z and Σ is the set of all even integers,
    • describe Φ(Σ) in the case when X = R and Σ = {0},
    This is called the indicator function. What are you having difficulty with? It's evident what \Phi(\varnothing) (as you put it) is, right? \displaystyle \left(\Phi(\varnothing)\right)(x)=\begin{cases}1 & \mbox{if}\quad x\in\varnothing\\ 0 & \mbox{if}\quad x\notin\varnothing\end{cases}=\cdots=0.
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  3. #3
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    Oh, thanks a lot! I was confused because I missed the lecture...
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