1. **State the problem:** Find the complement of set $A=\{1,3,5,7\}$ with respect to the universal set $U=\{1,2,3,4,5,6,7\}$. 2. **Recall the definition:** The complement of $A$ with respect to $U$, denoted $A^c$, is the set of all elements in $U$ that are not in $A$. 3. **Apply the definition:** $$A^c = U - A = \{x \in U : x \notin A\}$$ 4. **List elements of $U$ not in $A$:** $U = \{1,2,3,4,5,6,7\}$ $A = \{1,3,5,7\}$ Elements not in $A$ are $2,4,6$. 5. **Therefore, the complement is:** $$A^c = \{2,4,6\}$$ --- 6. **State the problem:** Find $A \cup B$ where $A = \{x : x \in \mathbb{N}, x^2 \leq 10\}$ and $B = \{x : x \in \mathbb{Z}, x^2 \leq 10\}$. 7. **Identify sets:** - $\mathbb{N}$ (natural numbers) typically means $\{1,2,3,\ldots\}$ - $\mathbb{Z}$ (integers) means $\{\ldots,-2,-1,0,1,2,\ldots\}$ 8. **Find elements of $A$:** $x^2 \leq 10$ means $x \in \mathbb{N}$ and $x^2 \leq 10$. Possible $x$ are $1,2,3$ because $3^2=9 \leq 10$ but $4^2=16 > 10$. So, $$A = \{1,2,3\}$$ 9. **Find elements of $B$:** $x \in \mathbb{Z}$ and $x^2 \leq 10$. Possible $x$ are $-3,-2,-1,0,1,2,3$ because $(-3)^2=9 \leq 10$ and $4^2=16 > 10$. So, $$B = \{-3,-2,-1,0,1,2,3\}$$ 10. **Find union $A \cup B$:** $$A \cup B = \{-3,-2,-1,0,1,2,3\} = B$$ **Final answers:** - Complement of $A$ with respect to $U$ is $\{2,4,6\}$. - $A \cup B = B$.