# Induction proof problem (for set union)

• deviltaz
In summary, the conversation is about a problem involving a proof by induction for a set equation. The speaker is unsure of how to use the induction hypothesis and asks for help. Another person suggests using complete induction and offers a suggestion for how to approach the problem. The original speaker clarifies a typo and explains the reasoning behind the equation. The conversation ends with a final suggestion to try the previously given advice.
deviltaz
I hope that someone can help me with the following problem:

Problem: Proof by induction that:

A1 $$\cup$$ A2 $$\cup$$...$$\cup$$An=(A1-A2)$$\cup$$(A2-A3)$$\cup$$...$$\cup$$(An-1-An)$$\cup$$(An-A1)$$\cup$$
(A1$$\cap$$A2$$\cap$$...$$\cap$$An)

Last edited:
Can you attempt the problem first? Or at least show us your work thus far.

If you don't I am not sure you will get any help :-).

The problem is that I don’t have any idea how to use the ‘induction hypothesis’.
(I can use it If I add the set An+1 to the left side of the equation, but this leads me to nothing).
The hypothesis somehow should be used (I guess) to the right side of the equation (using set algebra properties, but I don’t know how…)

Have you heard of complete induction ?

Basically you want your base case to consist of A_1 and A_2. Prove equality for those cases and form your induction hypothesis by assuming the inequality is true for all m<n. Then show the inequality holds for n.

EDIT
Can you check your post for typos?

I see a term An-A1. Is that supposed to be there ?

Last edited:
Yes, its suppose to be there - that is the basic idea of the equation (of the sets union) - adding the ‘next set’ does not take into consideration the elements of the first one and the ‘previous’ does not take into consideration the elements of the actual set.

What I meant is that for the An+1 set, the equation would takes the following form:

A1 $$\cup$$ A2 $$\cup$$...$$\cup$$An$$\cup$$An+1=(A1-A2)$$\cup$$(A2-A3)$$\cup$$...$$\cup$$(An-1-An)$$\cup$$(An-An+1)$$\cup$$(An+1-A1)$$\cup$$(A1$$\cap$$A2$$\cap$$...$$\cap$$An$$\cap$$An+1)

Try the suggestion I gave you. :-)

## What is an induction proof problem for set union?

An induction proof problem for set union is a mathematical proof technique used to show that a statement is true for all elements in a set by first proving it is true for the base case, and then showing that if it is true for any one element, it must also be true for the next element in the set.

## What is the base case in an induction proof for set union?

The base case in an induction proof for set union is the initial element or set of elements that the statement is proven to be true for. This is typically the smallest or simplest element in the set.

## What is the inductive step in an induction proof for set union?

The inductive step in an induction proof for set union is the process of showing that if the statement is true for one element in the set, it must also be true for the next element in the set. This is done using mathematical reasoning and the properties of set union.

## What are the key points to consider when constructing an induction proof for set union?

When constructing an induction proof for set union, it is important to consider the base case, the inductive step, and the overall structure of the proof. Additionally, it is important to ensure that the statement being proved is well-defined and that the properties of set union are used correctly.

## What are some common mistakes to avoid when using induction for set union?

Some common mistakes to avoid when using induction for set union include incorrectly identifying the base case, assuming the statement is true without properly showing the inductive step, and using properties of set union that do not hold true for all sets. It is also important to avoid circular reasoning and to clearly explain each step of the proof.

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