C_U (M \cap N)

Key takeaway: This question uses the complement of an intersection inside a universal set. The key is to find the common elements of $M$ and $N$, then list everything in $U$ that is not in that overlap.

Step 1: Find the intersection

Given:

• $U = \{1,2,3,4,5,6,7,8\}$
• $M = \{3,4,5,6\}$
• $N = \{4,5,6,7\}$

The intersection is the set of elements common to both $M$ and $N$:

$M \cap N = \{4,5,6\}$

Step 2: Take the complement in $U$

The complement of $M \cap N$ in $U$ is everything in $U$ except $\{4,5,6\}$:

$C_U(M \cap N) = \{1,2,3,7,8\}$

Step 3: Match with the options

The correct complement is:

• $\{1,2,3,7,8\}$

If the options are exactly as written, none of the listed choices matches this result. The set corresponding to the correct computation is $\{1,2,3,7,8\}$.

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Pitfalls the pros know 👇 A common mistake is to complement $M$ and $N$ first and then intersect them, which gives a different result. Here the complement is taken after finding $M \cap N$.

Also, be careful not to exclude the wrong elements from the universal set $U$. Only $4,5,6$ belong to the intersection, so only those three should be removed.

What if the problem changes? If the question had asked for $C_U(M \cup N)$ instead, you would first compute

$M \cup N = \{3,4,5,6,7\}$

and then take the complement in $U$:

$C_U(M \cup N) = \{1,2,8\}$

That shows how changing the operation from intersection to union changes the answer.

Tags: intersection, complement, universal set