Proof on functions of an intersection of sets

In summary, when working with an arbitrary function g from the real numbers to the real numbers, it is always true that the image of the intersection of two sets A and B is a subset of the intersection of the images of A and B. To prove this, one can use the classical method of showing X is a subset of Y by picking an arbitrary value in X and using the definitions and properties of the sets to show it is also in Y. In this case, if a is in the image of the intersection of A and B, then it can be represented as g(x_0) for some value x_0 in the intersection of A and B. Using this information, one can show that a is also in the intersection of
  • #1
jeffreydk
135
0
I'm working out of Abbott's Understanding Analysis and I'm trying to show the following,

For an arbitrary function [itex]g :\mathbb{R}\longrightarrow \mathbb{R}[/itex] it is always true that [itex]g(A\bigcap B) \subseteq g(A) \bigcap g(B)[/itex] for all sets [itex]A, B \subseteq \mathbb{R}[/itex].

I'm confused on how to get going with this--any help or hints would be greatly appreciated. Thanks.
 
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  • #2
If [tex] X, Y [/tex] are sets for which you know the definitions or other properties, the classical way to show [tex] X \subset Y [/tex] is this:

1: Pick an arbitrary [tex] a \in X [/tex]

2: Use the definitions of the sets to show that [tex] a \in Y [/tex]

As a start, if you know that [tex] a \in g(A \cap B)[/tex], then you know that there is a value [tex] x_0 [/tex] such that [tex] a = g(x_0) [/tex] and that [tex] x_0 \in A \cap B [/tex]. What else do you know about [tex] x_0 [/tex], and how can you use that information?
 
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  • #3
Thanks, that really helps, I think I've got it now.
 

What is the definition of an intersection of sets?

An intersection of sets is a mathematical operation that results in a new set containing only the elements that are common to all of the original sets.

How is an intersection of sets represented?

An intersection of sets is typically represented using the symbol ∩.

What is the significance of studying functions of an intersection of sets?

Studying functions of an intersection of sets allows us to understand how elements in different sets are related and how they interact with each other.

How do you prove the existence of a function of an intersection of sets?

To prove the existence of a function of an intersection of sets, we must show that for every input in the intersection, there is a unique output.

What are some real-life applications of functions of an intersection of sets?

Functions of an intersection of sets are commonly used in statistics, probability, and data analysis to determine the likelihood of overlapping events or relationships between different sets of data.

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