Real analysis - unions and intersections

In summary, the conversation is discussing four problems related to sets and functions. The first two problems involve proving or disproving the equality of two expressions involving the function f, while the last two involve the inverse function g. The conversation also touches on the difference between f having an inverse and asking about f inverse of sets. The discussion concludes with a suggestion to try a simple counterexample to prove the reverse direction in problem 2.
  • #1
badsis
5
0
Hi,

I have four similar problems that I am not sure how to do: Given: A1 and A2 are in X, B1 and B2 are in Y f: X->Y, g - inverse of f
I have to either prove or if false find counterargument
1. f(A1 U A2) = f(A1) U f(A2)
2. f(A1 n A2) = f(A1) n f(A2)
3. g(-1)(B1 U B2) = g(B1) U g(B2)
4. g(B1 n B2) = g(B1) n f(B2)

I started doing 2. I was able to show that f(A1 n A2) C=(is contained in) f(A1) n f(A2):
let x € f(A1) and x € f(A2)
since (A1 n A2) <=A1, x€f(A1)
since (A1 n A2) <=A2, x€f(A2)
=> x € f(A1 n A2), x € f(A1) n f(A2), i.e. (A1 n A2) C= f(A1) n f(A2)

But I am not sure how to show the other way, i.e. that f(A1) n f(A2) C= (A1 n A2), in order to conclude that both expressions are equal. Or are they equal at all?
 
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  • #2
Are you saying that you are given that f has an inverse- that f is one-to-one and onto- or are you saying that (3) and (4) ask about f-1 of the sets? Those are very different things!
 
  • #3
This is set theory, not real analysis. The first one should be straightforward. Your notation is sort of hard to follow for 2, but that is the correct containment direction you concluded. Can you try coming up with a simple counterexample for the reverse direction? Consider perhaps a set A containing distinct elements a1 and a2, and let A1 = {a1} and A2 = {a2} and think of the simplest functions possible.
 
  • #4
Ok,
That worked.

Thanks!
 

FAQ: Real analysis - unions and intersections

1. What is the definition of a union in real analysis?

In real analysis, a union refers to the combining of two or more sets to create a new set that contains all the elements of the original sets.

2. How is the union of intervals represented in real analysis?

In real analysis, the union of intervals is represented by using the notation of "∪" or "U". For example, the union of the intervals [1,3] and [2,4] would be written as [1,3] ∪ [2,4].

3. What is the purpose of using unions in real analysis?

The purpose of using unions in real analysis is to create a new set that contains all the elements of the original sets, which can help in proving certain mathematical theorems and properties.

4. What is an intersection in real analysis?

In real analysis, an intersection refers to the common elements between two or more sets. It is represented by using the notation of "∩" or "n". For example, the intersection of the sets {1,2,3} and {2,3,4} would be written as {2,3} or {2,3} ∩ {2,3,4}.

5. How is the De Morgan's Law applied to unions and intersections in real analysis?

In real analysis, De Morgan's Law states that the complement of the union of two sets is equal to the intersection of their complements, and vice versa. This can be represented as (A ∪ B)^c = A^c ∩ B^c and (A ∩ B)^c = A^c ∪ B^c, where A and B are sets in real analysis.

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