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  • MAT102 Final Exam Prep

MAT102 Worksheet 2 Review

Q1

• Suppose that $A,B,C$ are sets, show that $A\setminus (B\cup C)=(A\setminus B)\cap (A\setminus C)$
• Proceed by double subset inclusion
• Let $x\in (A\setminus (B\cup C))$
• This means that $x\in A$ but $x\notin (B\cup C)$
• So then $x\in A$ but $x\notin B$ and $x\notin C$
• So then $x\in (A\setminus B)$ and $x\in (A\setminus C)$
• So $x\in (A\setminus B)\cap (A\setminus C)$
• $◻$

Q2

• Define a new logical operator $\otimes$ as $A\otimes B=\mathrm{\neg }(A\wedge B)$

A

• Construct the truth table for $A\otimes A$ and confirm that $\mathrm{\neg }A=A\otimes A$

B

• Construct the truth table for $(A\otimes B)\otimes (A\otimes B)$ and confirm this is equvalent to $A\wedge B$.

C

• Determine how to write $A\vee B$ in terms of onyl $A,B$ and $\otimes$

$\mathrm{\neg }(A\wedge B)=A\vee B$
$\mathrm{\neg }A\vee \mathrm{\neg }B=A\vee B$

$(A\otimes B)$

$A\vee B=\mathrm{\neg }(\mathrm{\neg }A\wedge \mathrm{\neg }B)=(A\otimes A)\otimes (B\otimes B)$