## PROBLEM 4

Given the sets: $A=\left\{x \in N^{*} \mid 2^{*} x \leq 8\right\}$,

$$
\begin{aligned}
& B=\left\{x \in N^{*} \mid x=2^{n-1}, n \in A\right\}, \\
& C=\{x \in N \mid x=m-n, m \in B, n \in A, m>n\} .
\end{aligned}
$$

a) Determine the elements of the sets $A, B$ and $C$;

b) Determine the elements of the set $A \cap B \cap C$;

c) Determine the truth value of the statement: "The set $(A \cup B \cup C) \backslash(A \cap B \cap C)$ contains only consecutive numbers."
## Solution:

$2 \cdot \mathrm{x} \leq 8$ and $\mathrm{x} \in \mathrm{N}^{*} \Rightarrow \mathrm{A}=\{1,2,3,4\}$ $\qquad$ 1 point
$\mathrm{x}=2^{\mathrm{n}-1}, \mathrm{n} \in \mathrm{A} \Rightarrow \mathrm{B}=\{1,2,4,8\}$ $\qquad$ 1 point
$x=m-n, m \in B, n \in A, m>n \Rightarrow C=\{1,2,3,4,5,6,7\}$ $\qquad$ 1 point
$\mathrm{A} \cap \mathrm{B} \cap \mathrm{C}=\{1,2,4\}$ $\qquad$ 1 point
$\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}=\{1,2,3,4,5,6,7,8\}$ 1 point
$(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}) \backslash(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=\{3,5,6,7,8\}$ $\qquad$ 1 point