MAT102 Lecture 3

• There is no smallest positive real number

  • Why?
    • There's always a smaller number, you can keep adding zeros to something like $0.000\dots 1$
  • The smallest positive real number is $r$.
  • There is an upper bound for the domain in which $r$ exists. From $0<r<s$
  • We create a variable $m=\frac{r}{2}$. Now $0<m<r<s$. Which makes no sense, because $r$ was supposed to be the smallest number, now it's $m$.

• We have sets A, B and C. Show that $A\text{}(B\cup C)=(A\text{}B)\cap (A\text{}C)$

• Let $x\in A$ so then $x\notin (B\cup C)$

  • $x\notin B$
  • $x\notin C$

• $x\in (A\setminus B)$

• $x\in (A\setminus C)$

• We know this because $x\notin B,x\notin C$

• If $x\in A$ then it must be part of $x\in (A\setminus B)\cap (A\setminus C)$

• We know $x\in A$ but $x\notin B$ and $x\notin C$
• So $x\notin (B\cup C)$
• We still have $x\in A$
• So $x\in A\setminus (B\cup C)$
• Left side = Right side