# Basic set theory question about complement

• damightytom
In summary, the conversation is about simplifying a set using set theory laws. Specifically, the set A U (A U B^c)^c intersect (A U C ) is simplified to AU(BnC) using De Morgans and Distributive Law. The conversation also discusses the relationships between relative and absolute complements.
damightytom

## Homework Statement

Hi I could use some help getting an explanation that kinda twists my head a bit.
I want to know what I'm missunderstanding so I can get this right from the beginning.
^c = complement
U = union

I want to simplify this set
A U (A U B^c)^c intersect (A U C )
to
AU(BnC)
Using set theory laws

## The Attempt at a Solution

So I start using set theory rules.

De Morgans
A U (A^c intersect B) intersect (A U C)

Distributive Law
A U (A^c intersect B) intersect C)

This seem to be the same as AU(BnC), so it seems A^c doesn't have any impact.
When using the distributive law can I do like this?
A U ((B^c)^c intersect C)
I would really appreciate any explanation on why this is and if there's any fault in my logic.
Thanks

damightytom said:

## Homework Statement

Hi I could use some help getting an explanation that kinda twists my head a bit.
I want to know what I'm missunderstanding so I can get this right from the beginning.
^c = complement
U = union

I want to simplify this set
A U (A U B^c)^c intersect (A U C )
to
AU(BnC)
Using set theory laws

## The Attempt at a Solution

So I start using set theory rules.

De Morgans
A U (A^c intersect B) intersect (A U C)

Distributive Law
A U (A^c intersect B) intersect C)

This seem to be the same as AU(BnC), so it seems A^c doesn't have any impact.
When using the distributive law can I do like this?
A U ((B^c)^c intersect C)
I would really appreciate any explanation on why this is and if there's any fault in my logic.
Thanks

you know that A^cB <=> A Intersect B barre*
Relationships between relative and absolute complements:
A ∖ B = A ∩ Bc (Bc is the elements out of the set b )
*sorry we read math in french don't know how to explain it it's b with a line
at Relationships between relative and absolute complements:

Last edited by a moderator:

## 1. What is a complement in set theory?

A complement in set theory is the set of all elements that are not included in a given set. It is denoted by a superscript 'c' after the set, for example, Ac.

## 2. How do you find the complement of a set?

To find the complement of a set, you need to first identify the universal set, which is the set of all elements under consideration. Then, you can find the complement by subtracting the given set from the universal set.

## 3. Can a set and its complement have any elements in common?

No, a set and its complement cannot have any elements in common. The complement of a set contains all the elements that are not in the original set, so they cannot overlap.

## 4. What is the relationship between a set and its complement?

The complement of a set is the opposite or negation of that set. It includes all the elements that are not in the original set. The two sets are mutually exclusive and together make up the entire universal set.

## 5. Are there any special properties of complements in set theory?

Yes, there are several properties of complements in set theory. Some of the most important ones include: the complement of the empty set is the universal set, the complement of the universal set is the empty set, the complement of the complement of a set is the set itself, and De Morgan's laws, which state that the complement of the union of two sets is equal to the intersection of their complements, and the complement of the intersection of two sets is equal to the union of their complements.

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