Given that the universal set $U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, we are tasked with finding the complement of the set $A = \{ x \mid x \in \mathbb{N}, x < 7 \}$.

Step 1: Define Set $A$

The set $A$ is defined as the natural numbers $x$ such that $x < 7$ and belongs to the universal set $U$. So, the set $A$ is: $A = \{0, 1, 2, 3, 4, 5, 6\}$

Step 2: Find the Complement of $A$

The complement of a set $A$, denoted $A^c$, consists of all elements in the universal set $U$ that are not in $A$. Therefore, we subtract the elements of $A$ from $U$: $A^c = U - A = \{7, 8, 9, 10\}$

Final Answer:

The complement of the set $A$ is: $\{7, 8, 9, 10\}$

Would you like further explanation or have any questions?

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Related questions:

1. What is the complement of a set with respect to a different universal set?
2. How does set complement work in infinite sets?
3. How can set complements be applied in probability?
4. What is the relationship between the union of a set and its complement?
5. How does De Morgan’s law apply to complements of sets?

Tip: When finding the complement, always ensure you know the elements in the universal set to avoid errors.