MAT102 Lecture 03B

• Proof:
  • $A\setminus (B\setminus C)\ne (A\setminus B)\cup C$
  • Let $\begin{array}{c}A=\{1,2\}\\ B=\{2,3\}\\ C=\{3,4\}\end{array}$
  • Then $A\setminus (B\setminus C)=\{1\}$ and $(A\setminus B)\cup C=\{1,3,4\}$
  • So these are not equal.
• Q3:
  • $A=\{n\in \mathbb{Z}:n^2<30\}$
  • $B=\{q\in \mathbb{Q}:2q\in \mathbb{Z}\}$
  • Notice that $(-5,\frac{2}{3})\notin B$ as if $\frac{2}{3}=q$ and $2q\in \mathbb{Z}$. Notice that $2\left(\frac{2}{3}\right)=\frac{4}{3}$ and $\frac{4}{3}\notin \mathbb{Z}$.
• Q4:
  • P True
  • Q False
  • R True
• Q5:
  • $\begin{array}{cccc}A& B& \mathrm{\neg }A\vee B& (\mathrm{\neg }(\mathrm{\neg }A\vee B))\\ T& T& T& F\\ T& F& F& T\\ F& T& T& F\\ F& F& T& F\end{array}$