Year 12 Set Theory: Set operations

Set operations are the basic language of the topic. Once you know what union, intersection, complement and difference mean, most set questions become translation exercises.

Translate words into symbols as early as possible. "Or" usually means union, "and" usually means intersection, and "not" usually points to a complement.

Main operations

$A \cup B$ : elements in A or B or both.
$A \cap B$ : elements that are in both A and B.
$A'$ or $A^c$ : elements not in A.
$A \ B$ : elements in A but not in B.

Worked example

Problem: Let

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {2, 4, 6, 8, 10}

and

B = {2, 3, 5, 7}

. Find

A \cup B

A \cap B

A'

and

A \ B

For $A \cup B$ , collect everything that is in A or B: ${2, 3, 4, 5, 6, 7, 8, 10}$ .
For $A \cap B$ , keep only the common elements: ${2}$ .
For $A'$ , list everything in U that is not in A: ${1, 3, 5, 7, 9}$ .
For $A \ B$ , keep elements in A but remove anything also in B: ${4, 6, 8, 10}$ .

Answer:

A \cup B = {2, 3, 4, 5, 6, 7, 8, 10}

A \cap B = {2}

A' = {1, 3, 5, 7, 9}

and

A \ B = {4, 6, 8, 10}

What to watch

Do not confuse $A \ B$ with $B \ A$ .
Complements depend on the universal set.
Use brackets carefully when several operations appear in one line.

Worked example 2

Problem: Let

U = {1, 2, 3, 4, 5, 6}

A = {1, 2, 3, 4}

and

B = {3, 4, 5}

. Find

(A \cup B)'

First find the union: $A \cup B = {1, 2, 3, 4, 5}$ .
Now take the complement in the universal set $U$ .
The only element of $U$ not in the union is 6.

Answer:

(A \cup B)' = {6}

Why notation order matters

In a multi-step expression such as $(A \cup B)'$ or $A \ (B \cap C)$ , the brackets tell you which operation happens first. Without respecting that order, you can produce a set with a completely different meaning.

Revision focus

Set theory rewards precise translation. A phrase such as in A but not B, in both sets, in at least one set, or outside the union should immediately become a symbolic expression. Once the wording has been translated cleanly, most problems reduce to region tracking, addition rules, or careful interpretation of complements and intervals. Students usually struggle not because the arithmetic is hard, but because the language was never converted into symbols with enough care.

A very effective checking strategy is to compare the completed diagram or set statement against the universal set. Do all regions add back to the total? Does the complement really refer to everything outside the chosen set? If an interval is being described, does it contain every value between its endpoints? These checks are simple, but they are exactly the sort of checks that stop double counting and notation errors from surviving into the final answer.