Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find A – B and B – A

Question

Accepted Answer

> [!TLDR]
> Set difference $A - B$ keeps everything in $A$ that is not in $B$; $B - A$ keeps everything in $B$ that is not in $A$. For the given sets, $A - B = \{1, 3, 5\}$ and $B - A = \{8\}$.

The idea behind set difference is simple: you start with one set and remove any elements that also appear in the other set. Think of it like a filter — only the elements that belong exclusively to the first set survive.

For $A - B$, we take set $A$ and delete every element that also lives in $B$. For $B - A$, we do the reverse: start with $B$ and remove anything that is also in $A$.

Let’s work through it step by step.

1. **List the elements of $A$ and $B$ clearly.**  
   $A = \{1, 2, 3, 4, 5, 6\}$  
   $B = \{2, 4, 6, 8\}$

2. **Find $A - B$.**  
   Go through each element of $A$:
   - $1$ is in $A$ but not in $B$ → keep it.
   - $2$ is in $A$ and also in $B$ → remove it.
   - $3$ is in $A$ but not in $B$ → keep it.
   - $4$ is in $A$ and also in $B$ → remove it.
   - $5$ is in $A$ but not in $B$ → keep it.
   - $6$ is in $A$ and also in $B$ → remove it.  
   So the survivors are $\{1, 3, 5\}$.  
   Hence $A - B = \{1, 3, 5\}$.

3. **Find $B - A$.**  
   Now go through each element of $B$:
   - $2$ is in $B$ and also in $A$ → remove it.
   - $4$ is in $B$ and also in $A$ → remove it.
   - $6$ is in $B$ and also in $A$ → remove it.
   - $8$ is in $B$ but not in $A$ → keep it.  
   Only $8$ remains.  
   Hence $B - A = \{8\}$.

> [!WARNING]
> A common mistake is to think $A - B$ and $B - A$ are the same thing, or that they always have the same number of elements. They are completely different sets — $A - B$ removes elements of $B$ from $A$, while $B - A$ removes elements of $A$ from $B$. Here, $A - B$ has three elements and $B - A$ has just one.

> [!TIP]
> Notice that $A - B$ and $B - A$ are always disjoint (they share no elements). Also, the union $(A - B) \cup (B - A)$ is called the *symmetric difference* of $A$ and $B$, often written $A 	riangle B$. In this problem, $A

Property	Statement
Not commutative	$A - B \neq B - A$ in general
Difference with itself	$A - A = \emptyset$
Difference with empty set	$A - \emptyset = A$ , and $\emptyset - A = \emptyset$
Difference with universal set	$U - A = A^c$ (the complement of $A$ )
Disjoint sets	If $A \cap B = \emptyset$ , then $A - B = A$

Concept	Why it's defined this way
Set	To have a precise, unambiguous collection — no guesswork.
Set-builder	To define infinite or complex sets without listing.
Membership ( $\in$ )	The only question that matters — is it inside or not?
Empty set	Logical completeness — a property may have no objects.

Step	What to do
1	Write down the first set completely
2	Cross out any element that also appears in the second set
3	List the remaining elements once each
4	Double-check: Did you accidentally add anything from the second set?

Q.Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find A – B and B – A

Concept understanding — Set Difference

Set Difference

The idea in plain words

The precise definition

Worked example

Key properties

Set difference vs. complement — the classic mix-up

Set difference vs. symmetric difference

Why it matters

Takeaway

Why this formula?

1. What is a Set? (The Core Idea)

2. The Key "Formula": Set-Builder Notation

Why does this work?

3. The Two Fundamental Properties (Axioms)

(a) Extensionality — Two sets are equal if they have the same elements.

(b) Membership — The only relation is $\in$ (belongs to).

4. Why Can't We Just List Everything?

5. The "Empty Set" — Why It Exists

6. Summary: The "Why" Behind the Definition

Method: Set Difference (Subtraction) Method

Concept First

Steps for $A - B$

Steps for $B - A$

Key Exam Tip

Common Mistakes in Set Membership & Set Difference

Mistake 1: Confusing the Order of Subtraction

Mistake 2: Including Elements from the Second Set That Aren't in the First

Mistake 3: Forgetting That Repetition Doesn't Matter

Mistake 4: Confusing Set Difference with Complement

Quick Checklist to Avoid Mistakes

Q.Let A = { 1, 2, 3, 4, 5, 6}, B = { 2, 4, 6, 8 }. Find A – B and B – A

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