Year 12 Set Theory: The addition rule

The addition rule tells you how to count the union of two sets without double counting the overlap. It is a set-theory rule first, and later the same structure appears in probability.

Add the two totals, then subtract the intersection once:

n (A \cup B) = n (A) + n (B) - n (A \cap B)

Why subtraction is needed

If you add $n (A)$ and $n (B)$ directly, every element in $A \cap B$ gets counted twice. Subtracting the intersection once fixes that.

Worked example

Problem: In a class of 40 students, 23 play tennis, 18 play soccer and 7 play both. How many students play at least one of the two sports?

Use the addition rule: $n (T \cup S) = n (T) + n (S) - n (T \cap S)$ .
Substitute the values: $n (T \cup S) = 23 + 18 - 7$ .
Calculate: $n (T \cup S) = 34$ .

Answer: 34 students play at least one of tennis or soccer.

Connection to probability

The same pattern appears in probability: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ . The reasoning is identical because the overlap would otherwise be counted twice.

Worked example 2

Problem: In a survey, 19 students like chess, 15 like coding and 6 like both. Find the number who like chess or coding.

Write the addition rule: $n (C \cup D) = n (C) + n (D) - n (C \cap D)$ .
Substitute: $19 + 15 - 6 = 28$ .

Answer: 28 students like chess or coding.

Common mistakes

Adding the two totals and forgetting to subtract the overlap.
Subtracting the overlap twice.
Using the rule when the sets are not the two being asked about.

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.

A useful closing check in set theory is to compare every region or notation statement against the wording of the problem one more time. If the language says exactly one, at least one, neither, or outside, the final symbolic answer should reflect that language precisely.

In short, the addition rule is one of the clearest examples of why overlap cannot be ignored. Counting accurately means deciding exactly which elements have already been included once.