Set theory: Is my proof valid?

[r0bHadz]

Homework Statement

Prove the following for a given universe U

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

Homework Equations

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) ≠ ∅

 

[Stephen Tashi]

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.

 

[vela]

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 $x \in B$.

 

[nuuskur]

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
$$X\lor Y \equiv \neg X\Rightarrow Y .$$

Spoiler: unnecessary

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.

 

[epenguin]

Homework Statement

Prove the following for a given universe U

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

Homework Equations

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.