#import "@youwen/zen:0.1.0": *
#import "@preview/ctheorems:1.1.3": *

#show: zen.with(
  title: "Homework 1",
  author: "Youwen Wu",
  date: "Winter 2025",
)

1. #[
    #set enum(numbering: "a)", spacing: 2em)

    + #[
        We know that $B$ and $B'$ are disjoint. That is, $B sect B' =
        emptyset$. Additionally,
        $
          E = (A sect B) subset B \
          F = (A sect B') subset B' \
        $
        Then we note
        $
          forall x in E, x in B, x in.not B' \
          forall y in F, y in B, y in.not B'
        $
        So clearly $E$ and $F$ have no common elements, and
        $
          E sect F = emptyset
        $
      ]

    + #[
        $
          E union F &= (A sect B) union (A sect B') \
          &=
        $
      ]
  ]