Recipe

Find a union, intersection, or difference

Use when: you're given two sets by their elements and asked for $A \cup B$ , $A \cap B$ , or $A \setminus B$

Goal: the exact resulting set, with no duplicate elements and the right direction if it's a difference

Taught in Union, intersection, difference

Quick recipe

List both sets' elements side by side
Identify which operation is being asked for, then apply its rule
For a difference, double-check the direction
Write the final answer in set notation

1
Write out $A$ and $B$ as element lists next to each other, before combining anything. Seeing both lists together makes every later comparison direct.

Example

$A = \{1,2,3,4\}$ and $B = \{3,4,5,6\}$ , written side by side so each element's membership in $A$ , in $B$ , or in both is easy to check at a glance.

Why: comparing elements is easiest when both lists sit in front of you, not buried in set-builder notation.
2
Union, intersection, and difference each keep a different subset of the elements you just listed, so match the question to the right rule.

Example

For $A = \{1,2,3,4\}$ , $B = \{3,4,5,6\}$ : union keeps $1,2,3,4$ from $A$ plus the new $5,6$ from $B$ . Intersection keeps only $3$ and $4$ , the elements both lists share. Difference $A \setminus B$ removes $B$ 's elements ( $3,4,5,6$ ) from $A$ , leaving $1$ and $2$ .
- union ( $A \cup B$ ) — keep every element appearing in A or in B, writing each one only once.
- intersection ( $A \cap B$ ) — keep only elements appearing in both A and B.
- difference ( $A \setminus B$ ) — keep only A's elements that do not also appear in B.
Why: each operation matches a different logical rule (or, and, and-not), so the right rule has to match the question asked, not just the two sets involved.
3
$A \setminus B$ and $B \setminus A$ generally name different sets, since one starts from A's elements and the other starts from B's.

Example

With $A = \{1,2\}$ , $B = \{2,3\}$ : $A \setminus B = \{1\}$ , but $B \setminus A = \{3\}$ . Swapping the two sets changes the answer.
- asked for $A \setminus B$ — remove B's elements from A's list.
- asked for $B \setminus A$ — remove A's elements from B's list.
Why: swapping which set you start from changes which elements get removed, so difference is not symmetric the way union and intersection are.
4
Collect the surviving elements inside braces, dropping any repeat, since a set never lists the same element twice.

Why: this confirms the result is itself a properly written set, matching the notation the question used.

Common mistakes

Treating $A \setminus B$ and $B \setminus A$ as if they always name the same set.
Listing a shared element twice when writing the union instead of once.
Confusing intersection's "and" with union's "or," swapping which elements survive.
Stopping after finding the union or intersection when the question actually asked for all three.