Proving "[a]_n ∩ [b]_n is Empty or Equals [b]_n

In summary, the problem is to prove that either the intersection of [a]_{n} and [b]_{n} is an empty set, or [a]_{n} is equal to [b]_{n}. The attempt at a solution involves assuming there is an element x in the intersection and showing that this implies [a]_{n}=[b]_{n}. The conversation then discusses the conditions for x being in both [a]_{n} and [b]_{n}, and the difficulty in determining the relationship between a and b in this case.
  • #1
kathrynag
598
0

Homework Statement


Prove that either [tex][a]_{n}[/tex][tex]\cap[/tex][tex]_{n}[/tex]=empty set or [tex][a]_{n}[/tex]=[tex]_{n}[/tex].

Homework Equations


The Attempt at a Solution


I want to assume there is an element x in [tex][a]_{n}[/tex][tex]\cap[/tex][tex]_{n}[/tex] and show this implies [tex][a]_{n}[/tex]=[tex]_{n}[/tex].
This tells me x is in [tex][a]_{n}[/tex] and [tex]_{n}[/tex].
That's where I get stuck.
 
Last edited:
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  • #2
sorry, I meant to show this implies [a]=.
 
  • #3
If x=a mod n then n|x-a. x=b mod n means n|x-b.

If x|x-a and n|x-b... what can you say about a-b?
 
  • #4
x-a=x-b
x-x=a-b
0=a-b
b=a
 
  • #5
x-a is probably not equal to x-b
 
  • #6
Ok then I'm not really sure where to go with a-b then.
 

FAQ: Proving "[a]_n ∩ [b]_n is Empty or Equals [b]_n

1. What does it mean to prove that [a]_n ∩ [b]_n is empty or equals [b]_n?

To prove that [a]_n ∩ [b]_n is empty or equals [b]_n means to show that there is no overlap or intersection between the two sets. This can be proven by showing that the two sets have no common elements or that one set is entirely contained within the other.

2. Why is it important to prove that [a]_n ∩ [b]_n is empty or equals [b]_n?

Proving that [a]_n ∩ [b]_n is empty or equals [b]_n is important because it allows us to understand the relationship between the two sets. It can also help us to determine if the sets are mutually exclusive or if they have any common elements.

3. What are some methods for proving that [a]_n ∩ [b]_n is empty or equals [b]_n?

There are a few different methods for proving that [a]_n ∩ [b]_n is empty or equals [b]_n. One method is to use the definition of set intersection and show that there are no common elements between the two sets. Another method is to use proof by contradiction, assuming that there is an element that belongs to both sets and then showing that this leads to a contradiction. Additionally, using set identities and algebraic manipulation can also be used to prove that [a]_n ∩ [b]_n is empty or equals [b]_n.

4. Can [a]_n ∩ [b]_n be empty and equal [b]_n at the same time?

No, [a]_n ∩ [b]_n cannot be empty and equal [b]_n at the same time. If the intersection is empty, it means that there are no common elements between the two sets. If [a]_n ∩ [b]_n equals [b]_n, it means that all the elements in [b]_n are also in [a]_n. These two statements contradict each other, so they cannot both be true at the same time.

5. How can proving that [a]_n ∩ [b]_n is empty or equals [b]_n be applied in real life situations?

Proving that [a]_n ∩ [b]_n is empty or equals [b]_n can be applied in many different real-life situations. For example, it can be used in statistics to analyze data sets and determine if there are any common elements between two groups. It can also be applied in decision making, such as determining if two options are mutually exclusive or if they have any similarities. Additionally, it can be used in computer programming to check for duplicate values in different sets of data.

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