# How Can I Prove the Equality of Union and Intersection in This Homework?

• chocolatelover
In summary, the student is trying to solve a problem they don't understand, and is looking for help from others.
chocolatelover

## Homework Statement

Prove (A is a union of B)/(A is an intersection of B)=(A/B) is a union of (B/A)

## The Attempt at a Solution

Could someone first help me translate all of this into plain English. I don't really understand what I need to prove. Would I start off with the contrapositive? Is the contrapositive "If (A/B) is not the union of (B/A), then A is not the union of B/(A is not the intersection of B) and it is not equal to the antecedent"? Could someone please show me where to go from here?

Thank you very much

chocolatelover said:

## Homework Statement

Prove (A is a union of B)/(A is an intersection of B)=(A/B) is a union of (B/A)

## The Attempt at a Solution

Could someone first help me translate all of this into plain English. I don't really understand what I need to prove. Would I start off with the contrapositive? Is the contrapositive "If (A/B) is not the union of (B/A), then A is not the union of B/(A is not the intersection of B) and it is not equal to the antecedent"? Could someone please show me where to go from here?

Thank you very much

I assume the question was given as:
$$(A \cup B)/(A \cap B) = (B/A)\cup(A/B)?$$
If x is an element of the set on LHS then x is in A or x is in B but x is not in both A and B

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This is called the symmetric difference of two sets. It can be proven the the associative, distributive, and commutative laws holds with symmetric difference. Those are good exercises.

Thank you very much

I assume the question was given as:

That's correct, except A and B are switched in the second part. (A/B) U (B/A)

Would the contrapositive also prove it?

I know how to use the associative property, but I'm sure how how use the others to prove this. I know that, say, A upside B upside C=(A upside B) upside U C=A upside U (B upside U C) I'm not sure how to do that or the others for this problem. Would it be (A U B)/(A upside U B)=A U B/A upside U B?

Thank you

Last edited:

## What is a union?

A union is an operation that combines two sets, resulting in a new set that contains all the elements from both original sets without any duplicates.

## What is an intersection?

An intersection is an operation that combines two sets, resulting in a new set that only contains elements that are common to both original sets.

## How do you prove the union of two sets?

To prove the union of two sets, you need to show that every element in the resulting set is also in at least one of the original sets. This can be done by using the set builder notation or by using logical statements.

## How do you prove the intersection of two sets?

To prove the intersection of two sets, you need to show that every element in the resulting set is also in both of the original sets. This can also be done using set builder notation or logical statements.

## What is the difference between union and intersection?

The main difference between union and intersection is that union combines sets by including all elements from both sets, while intersection only includes elements that are common to both sets. Union can also result in a larger set, while intersection can result in a smaller set or even an empty set.

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