Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′

Question

Accepted Answer

> [!TLDR]
> The complement of a set is everything in the universal set that is not in the original set. Here, $A' = \{2, 4, 6, 8, 10\}$.

The idea of a set complement is simple but powerful: it’s the “other half” of the universe. If $U$ is the universal set (the collection of all elements we care about), then $A'$ (read “A complement” or “A prime”) contains every element of $U$ that is **not** in $A$.

Think of it like a Venn diagram: $U$ is the entire rectangle, $A$ is one circle inside it. $A'$ is everything outside that circle but still inside the rectangle. So to find $A'$, you just scan through $U$ and pick out the numbers that are missing from $A$.

Let’s do it step by step.

1. **List the universal set.**  
   $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$

2. **List the given set $A$.**  
   $A = \{1, 3, 5, 7, 9\}$

3. **Identify what’s missing.**  
   Compare each element of $U$ against $A$:
   - $1$ is in $A$ → skip
   - $2$ is not in $A$ → include
   - $3$ is in $A$ → skip
   - $4$ is not in $A$ → include
   - $5$ is in $A$ → skip
   - $6$ is not in $A$ → include
   - $7$ is in $A$ → skip
   - $8$ is not in $A$ → include
   - $9$ is in $A$ → skip
   - $10$ is not in $A$ → include

4. **Collect the complement.**  
   The elements we kept are $2, 4, 6, 8, 10$. So:
   $$A' = \{2, 4, 6, 8, 10\}$$

> [!WARNING]
> A common mistake is to forget that the complement is defined **only** with respect to the given universal set $U$. If $U$ were different, $A'$ would change. Here, $U$ is clearly given, so we stick to it.

> [!TIP]
> Notice that $A$ contains all the odd numbers from $U$, and $A'$ contains all the even numbers. That’s a neat pattern: complements often reveal a natural partition of the universal set.

> [!ANSWER]
> The complement of $A$ is $\boxed{\{2, 4, 6, 8, 10\}}$.

Operation	What it asks	Bucket contains
Union ( $A \cup B$ )	In either?	Everything from $A$ or $B$ (or both)
Intersection ( $A \cap B$ )	In both?	Only the overlap
Difference ( $A \setminus B$ )	In $A$ but not $B$ ?	Just the part of $A$ that doesn't touch $B$
Complement ( $A^c$ )	Not in $A$ ?	Everything outside $A$ (inside $U$ )
Symmetric Difference ( $A \triangle B$ )	In exactly one?	The two crescent-shaped parts, excluding the overlap

Step	Action
1	Write $U$ clearly.
2	Write $A$ clearly.
3	Cross out every element of $A$ from $U$ .
4	List the remaining elements in $\{ \}$ .
5	Verify: no element of $A$ is in your answer.
6	Verify: every element of your answer is in $U$ .

Q.Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′

Concept understanding — Set Operations

The idea in plain words

Why this works

Step by step

A common slip

Takeaway

Method: Set Difference (Subtraction) Method

Steps for $A - B$

Steps for $B - A$

Key Concept Check

🧠 The Concept: Complement of a Set

✗ Common Mistake #1: Forgetting the Universal Set

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

✗ Common Mistake #3: Confusing Complement with Difference

✗ Common Mistake #4: Writing $A'$ as a Sentence Instead of a Set

✓ Quick Checklist to Avoid All Mistakes

Q.Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}. Find A′

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