---
Created: 2025-09-25
Type: Zettel
aliases:
References:
Links:
- "[[Set Theory]]"
tags:
- MATH31AH
---
- $\cap$ : "[[Intersection]]", $A \cap B$ is the set of elements of both $A$ and $B$
> Proof: prove $E_{5} \cap E_{3}=E_{15}$ (where $E_{n}$ is the set of natural numbers divisible by $n$)
>If $E_{5} \cap E_{3} = E_{15}$, then two conditions must be true
> 1. $E_{5} \cap E_{3} = E_{15}$
> 2. $E_{15}=E_{5}\cap E_{3}$
> To prove the case 1, we have to pick some $n \in E_{5}\cap E_{3}$. We know that $n\in E_{5}$ and $n\in E_{15}$.
> We can write $n=5k_{1}$, where $k_{1}\in \mathbb{N}$ .
> Because we know that $n$ is also divisible by 3, we also know that there must be some $p$ such that $p=\frac{3}{5k_{1}}$
> We got to case 2, where we pick $p\in E_{15}$.
> $p=15k_{2}$ where $k_{2} \in \mathbb{N}$
> Then, $p=\frac{3}{5}k_{1}$
> (incomplete)
>
