1. **State the problem:** We have three sets: - Universal set $\xi = \{x : x \text{ is an integer and } 1 \leq x \leq 6\}$ - Set $A = \{x : x \text{ is an integer and } 7 < 3x < 15\}$ - Set $B = \{x : x \text{ is a root of } x^2 - 8x + 15 = 0\}$ We need to find the elements of $A$ and $B$ and then place all elements of $\xi$, $A$, and $B$ in a Venn diagram. 2. **Find elements of $A$:** Given $7 < 3x < 15$, divide all parts by 3: $$7 < 3x < 15 \implies \frac{7}{3} < x < 5$$ Since $x$ is an integer, $x$ can be $2, 3, 4$ because $\frac{7}{3} \approx 2.33$ and $x$ must be greater than that and less than 5. So, $A = \{3, 4\}$ (since 2 is not greater than 2.33, only 3 and 4 satisfy). 3. **Find elements of $B$:** Solve the quadratic equation: $$x^2 - 8x + 15 = 0$$ Factor: $$x^2 - 8x + 15 = (x - 3)(x - 5) = 0$$ So roots are $x = 3$ and $x = 5$. Thus, $B = \{3, 5\}$. 4. **Universal set $\xi$:** $\xi = \{1, 2, 3, 4, 5, 6\}$. 5. **Place elements in Venn diagram:** - Elements in $A$ only: $4$ (since 3 is also in $B$) - Elements in $B$ only: $5$ - Elements in both $A$ and $B$ (intersection): $3$ - Elements in $\xi$ but not in $A$ or $B$: $1, 2, 6$ **Final sets:** $$A = \{3, 4\}$$ $$B = \{3, 5\}$$ $$A \cap B = \{3\}$$ $$A - B = \{4\}$$ $$B - A = \{5\}$$ $$\xi - (A \cup B) = \{1, 2, 6\}$$