Q.Let U be universal set of all the students of Class XI of a coeducational school and A be the set of all girls in Class XI. Find A′

The complement of a set consists of all elements in the universal set that do not belong to the given set.

Given:

• $U$ = all students of Class XI
• $A$ = all girls in Class XI

Finding $A'$:

The complement $A'$ (also written as $A^c$ or $\overline{A}$) contains every student in Class XI who is not a girl. …

Understanding Set Complement

When we talk about the complement of a set, we're asking: what's left over? The complement $A'$ (also written $A^c$ or $\overline{A}$) consists of every element in the universal set $U$ that does not belong to $A$.

The universal set here is all students in Class XI of a coeducational school. Since it's coeducational, the school has both boys and girls. If $A$ contains all the girls, then naturally everything in $U$ that isn't in $A$ must be the boys.

Finding the Complement

1. Identify the universal set: $U = \{\text{all students of Class XI}\}$. This includes every single student in the class, regardless of gender.

2. Identify the given set: $A = \{\text{all girls in Class XI}\}$. This is a subset of $U$ containing only the female students.

3. Apply the complement definition: The complement $A'$ contains all elements of $U$ that are not in $A$. Since every student is either a girl or a boy (in a coeducational setting), removing all girls leaves us with all boys.

4. Write the result: $A' = \{\text{all boys in Class XI}\}$. …

Method: Finding the Complement of a Set

Method Name: Complement via Set Subtraction

Why This Works

The complement $A'$ of a set $A$ (relative to a universal set $U$) is "everything in $U$ that is NOT in $A$." Since $A$ and $A'$ together account for all of $U$ with no overlap, finding $A'$ is a subtraction problem: $A' = U - A$.

Steps

Step 1: Identify the universal set $U$

Find what "everything" means in this problem — the full population under discussion.

Step 2: Identify the given set $A$

Note exactly which elements, or which description of elements, belong to $A$.

Step 3: Describe what's left over …

🧠 The Concept: Complement of a Set

Given a universal set $U$ and a set $A$, the complement $A'$ (also written $A^c$ or $\overline{A}$) is:

All elements in $U$ that are NOT in $A$.

So for:

• $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$
• $A = \{1, 3, 5, 7, 9\}$

The correct answer is:

✗ Common Mistake #1: Forgetting the Universal Set

What students do:

They list $A'$ as "everything except $A$" — but they don't check what $U$ actually contains. They might write $\{2, 4, 6, 8\}$ (missing 10) or include numbers like 11 or 12.

Why it happens:

They treat "complement" as "opposite of $A$ in all numbers" instead of "opposite of $A$ within $U$."

✓ How to avoid:

Always write $U$ down before starting. Then cross out every element of $A$ from $U$. Whatever remains is $A'$.

✗ Common Mistake #2: Including Elements of $A$ in $A'$

What students do:

They write $A' = \{1, 2, 4, 6, 8, 10\}$ — accidentally keeping 1 (which is in $A$).

Why it happens:

Rushing or misreading the question. They might think "complement" means "add the missing ones" instead of "remove all of $A$."

✓ How to avoid:

After writing your answer, check every element: is it in $A$? If yes, it cannot be in $A'$. Double-check by scanning $A$ again.

✗ Common Mistake #3: Confusing Complement with Difference

What students do:

They write $A' = U - A$ correctly in concept, but then compute $U - A$ as $\{2, 4, 6, 8\}$ (forgetting 10) or $\{2, 4, 6, 8, 9\}$ (keeping 9).

Why it happens:

They don't systematically subtract — they guess or skip elements.

✓ How to avoid:

Use a systematic method:

1. List $U$ in order.
2. Circle or tick elements that are in $A$.
3. Write down only the unticked ones.

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