# Why is this property true of sets/indicator functions? (A page from my textbook)

• jdinatale
In summary, the conversation discusses the clarification of the underlined sentence in red. It is mentioned that ##E## is a measurable set and ##S_0(E)## is the set of simple functions on ##E##. The canonical representation of ##\phi \in S_0(E)## is also defined as a linear combination of characteristic functions, with additional conditions on the sets ##A_i## and ##B_j##. It is suggested that these additional conditions may explain the given equations.
jdinatale
I can't for the life of me figure out why the underlined sentence in red is true:

Could someone clarify?

That page doesn't tell us what ##E## or ##S_0(E)## represent. Nor what the canonical representation of ##\phi \in S_0(E)## is.

LCKurtz said:
That page doesn't tell us what ##E## or ##S_0(E)## represent. Nor what the canonical representation of ##\phi \in S_0(E)## is.

My apologies,

##E## is a measurable set.

##S_0(E)## is the set of simple functions on ##E##.

The canonical representation of ##\phi \in S_0(E)## is a linear combination of characteristic functions ##\phi = \sum_{i=1}^na_i\mathcal{X}_{E_i}##.

jdinatale said:
My apologies,

##E## is a measurable set.

##S_0(E)## is the set of simple functions on ##E##.

The canonical representation of ##\phi \in S_0(E)## is a linear combination of characteristic functions ##\phi = \sum_{i=1}^na_i\mathcal{X}_{E_i}##.

I'm guessing there is more to the definition of canonical representation. Maybe you are given that the ##A_i## are disjoint with each other and ##B_j## similarly. And if ##\cup_i A_i = E## and ##\cup_j B_j = E## that would explain those equations.

## 1. Why is it necessary to define indicator functions for sets?

Indicator functions, also known as characteristic functions, are necessary in order to represent sets in a mathematical and logical way. They help us determine whether an element is a member of a set or not, and allow us to perform operations on sets such as union, intersection, and complement. In other words, they provide a concise and precise way to express the properties of sets.

## 2. How are indicator functions used in set theory?

Indicator functions are used to represent sets as a function from a universal set to the set of 0s and 1s. The function takes a value of 1 if the element is a member of the set, and 0 if it is not. This allows us to perform operations on sets using logical operations on the corresponding indicator functions. For example, the intersection of two sets can be represented as the logical AND operation on their indicator functions.

## 3. Can indicator functions be used for infinite sets?

Yes, indicator functions can be used for infinite sets. In fact, they are particularly useful for infinite sets as they provide a way to represent and work with these sets in a finite and manageable manner. For example, the indicator function for the set of all positive integers would take a value of 1 for all positive integers and 0 for all other integers.

## 4. What is the relationship between indicator functions and characteristic equations?

Indicator functions and characteristic equations are essentially different ways of representing the same thing. The characteristic equation for a set is a logical statement that defines the set, while the indicator function is a function that represents the set using 0s and 1s. In set theory, the characteristic equation can be converted into an indicator function and vice versa.

## 5. How do indicator functions help in proving properties of sets?

Indicator functions are particularly useful in proving properties of sets as they allow us to use logical operations and mathematical techniques to manipulate and analyze sets. For example, we can use indicator functions to prove properties such as De Morgan's laws, which relate logical operations on sets. In addition, they provide a way to represent sets in a concise and rigorous manner, making it easier to analyze and prove properties about them.

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