Once you have two sets side by side, there are only two natural questions. Is it in this one OR that one? Is it in this one AND that one? The first builds the union, the second builds the intersection.
The four operations
| Name | Symbol | The question it asks |
|---|---|---|
| Union | A ∪ B | Is it in A OR in B, or in both? |
| Intersection | A ∩ B | Is it in A AND in B? |
| Difference | A − B | Is it in A but NOT in B? |
| Complement | A′ | Everything in the universal set that is not in A |
The union symbol is shaped like a cup, and a cup holds everything you pour into it. The intersection symbol is the same shape upside down, like a cap, and a cap only covers the part that both share.
The universal set
The universal set, written U, is the collection of all the objects under discussion. If U is the numbers 1 to 10 and A is the even ones, then A complement is {1, 3, 5, 7, 9}. The same set A can have completely different complements depending on what U is.
A question that asks for a complement must tell you the universal set somewhere. Read the whole question, underline U, and write it at the top of your working.
Counting a union without counting twice
If 18 play football, 12 play volleyball and 5 play both, the answer is not 30. The 5 who play both have been counted twice. Subtract the overlap once and you get 25.
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Every element of the intersection sits inside A and inside B, so adding the two totals counts it twice. Taking the intersection away once fixes exactly those elements and leaves the rest alone. It is a repair, not a formula to memorise.
Order matters for one of them
| Operation | Does the order matter? |
|---|---|
| A ∪ B | No. A ∪ B = B ∪ A |
| A ∩ B | No. A ∩ B = B ∩ A |
| A − B | YES, and this is a common trap |
If A = {1, 2, 3, 4} and B = {3, 4, 5}, then A − B = {1, 2} but B − A = {5}. Read A − B as what is left of A when B is taken out, and the direction becomes obvious.
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