# Thread: Group Theory - Describing Functions

1. ## 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},

2. Originally Posted by mbmstudent
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? $\displaystyle \left(\Phi(\varnothing)\right)(x)=\begin{cases}1 & \mbox{if}\quad x\in\varnothing\\ 0 & \mbox{if}\quad x\notin\varnothing\end{cases}=\cdots=0$.

3. Oh, thanks a lot! I was confused because I missed the lecture...