# Set theory: Is my proof valid?

In summary: If A∩(B compliment) = ∅, x∈A ∨ x∈ (B compliment), but not bothIf x∈A ∧ x∉(B compliment), then x∈B .

## Homework Statement

Prove the following for a given universe U

A⊆B if and only if A∩(B compliment) = ∅

## The Attempt at a Solution

Assume A,B, (B compliment) are not ∅
if A∩(B compliment) = ∅, x∈A ∨ x∈ (B compliment), but not both

If x∈A ∧ x∉(B compliment), then x∈B , because if they are in the same U and A∩(B compliment) = ∅ then A∩B must have a common element.

Also A⊆B because if A was outside of B, then A∩(B compliment) ≠ ∅

Your proof is not valid by the standards of a typical course in set theory. For example, you are using intuitive language such as "if A was outside of B" that has no formal definition.

You are not being specific about quantifiers. For example:

if A∩(B compliment) = ∅, x∈A ∨ x∈ (B compliment), but not both

You fail to quantify the variable "x". What you apparently mean is:

If ##A \cap B^c = \emptyset## then ##\forall x ( ( x \in A \lor x \in B^c) \land \lnot( x \in A \land x \in B^c))##.

You also fail to give a reason why that statement should be true. Apparently, you are relying on an intuitive picture of the situation. In elementary courses an intuitive argument may be acceptable.

FactChecker
You mean "complement," not "compliment."

Assume A,B, (B compliment) are not ∅
I'm not sure why you need this assumption.

if A∩(B compliment) = ∅, x∈A ∨ x∈ (B compliment), but not both
That's not necessarily true. x could be in neither A nor ##B^c##.

You need to prove two things.
1. If ##A \subset B##, then ##A \cap B^c = \emptyset##.
2. If ##A \cap B^c = \emptyset##, then ##A \subset B##.
For #2, for example, you would assume ##A \cap B^c = \emptyset##, then start with ##x \in A## and show that it logically leads to ##x \in B##.

FactChecker and PeroK
The law of excluded middle is healthy to know. For every subset $A\subseteq U$, where $U$ is some fixed universe and for every $x\in U$ it holds that $x\in A$ or $x\notin A$ (i.e $x\in A^c$). The result is immediate due to
$$X\lor Y \equiv \neg X\Rightarrow Y .$$
Alternatively one may prove by contradiction. For instance, prove the forward direction. Assume $A\subseteq B$ holds. Formally
$$\forall x\in U, x\in A\Rightarrow x\in B.$$
Now, assume for a contradiction $A\cap B^c\neq \emptyset$. Formally
$$\exists x\in U, x\in A\land x\notin B.$$
This is impossible since we assumed for every $x\in U$ the implication $x\in A\Rightarrow x\in B$ is true. So we have a contradiction, which we obtained by assuming $A\cap B^c\neq\emptyset$. This assumption must be false.

Last edited:

## Homework Statement

Prove the following for a given universe U

A⊆B if and only if A∩(B compliment) = ∅

## The Attempt at a Solution

Assume A,B, (B compliment) are not ∅
if A∩(B compliment) = ∅, x∈A ∨ x∈ (B compliment), but not both

If x∈A ∧ x∉(B compliment), then x∈B , because if they are in the same U and A∩(B compliment) = ∅ then A∩B must have a common element.

Also A⊆B because if A was outside of B, then A∩(B compliment) ≠ ∅

Learn on this occasion that the word is "complement".
It is related to the word complete.
Or think of a ship's complement - the number that needs to be made up to create a properly working crew.

Compliment is "a polite expression of praise or admiration".
Nothing logical – they both have the exact same Latin root, and originally even the same spelling in English - idea is a plain statement, answer, greeting, etc will often be regarded as by itself insufficient, needing to be complemented by a compliment.

## 1. What is set theory?

Set theory is a branch of mathematics that studies collections of objects, called sets, and the relationships between them. It provides a foundation for other areas of mathematics such as algebra and calculus.

## 2. How do I know if my proof in set theory is valid?

To determine if a proof in set theory is valid, you must first clearly define your axioms, assumptions, and logical rules. Then, you must present a clear and logical argument using these elements to prove your statement. Finally, you must check for any logical fallacies or errors in your reasoning.

## 3. Can I use diagrams or illustrations in my set theory proof?

Yes, diagrams and illustrations can be a helpful tool in understanding and presenting a proof in set theory. However, they should not be used as a replacement for clear and logical reasoning.

## 4. How can I improve my proof writing skills in set theory?

To improve your proof writing skills in set theory, it is important to familiarize yourself with the fundamental concepts and rules of the subject. Practice writing proofs and seek feedback from others, such as a mathematics professor or colleague. Reading and analyzing well-written proofs can also help improve your skills.

## 5. Are there any common mistakes to watch out for when writing a proof in set theory?

Some common mistakes to watch out for when writing a proof in set theory include using undefined or ambiguous terms, making incorrect assumptions, and failing to clearly explain your reasoning. It is also important to double check for any errors in your calculations or logical steps.

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