A large slice of the reasoning paper is arithmetic hidden inside language: how many people do at least one, how many do neither, how many do exactly one. Every one of those is a question about sets, and every one is solved by the same picture.
Venn diagrams also give you a second, independent way to test a syllogism. When "all", "no" and "some" start to blur, the overlapping circles answer the question visually - which region must be empty, and which must contain at least one item.
A set is just a collection. Draw each set as a circle and every item as a dot placed where it belongs.
What the model omits: the sizes. A Venn diagram shows which relationships are possible, but a counting question needs the actual number written into each region before you can answer.
| Symbol | Name | Means |
|---|---|---|
| A ∪ B | union | in A, in B, or in both |
| A ∩ B | intersection | in A and in B together |
| A′ | complement | not in A |
| n(A) | cardinality | the number of items in A |
| A × B | Cartesian product | all ordered pairs (a, b) with a in A and b in B |
A × B is the set of ordered pairs, first element from A, second from B. Its size is the product of the sizes: n(A × B) = n(A) × n(B), because every element of A pairs with every element of B. With A = {1, 2, 4} and B = {1, 3, 4, 5, 7}, n(A × B) = 3 × 5 = 15. Order matters — (1, 3) and (3, 1) are different pairs — so A × B and B × A are different sets whenever A ≠ B, even though they have the same size.
The subtraction is the whole idea: the items in both circles were counted once in A and once in B, so they must be removed once. For three sets the pattern alternates:
| Phrase | Region |
|---|---|
| at least one of the sets | the union |
| neither | outside the union - the complement |
| both A and B | the intersection A ∩ B |
| exactly one | the two single-set regions only, not the overlap |
In a class of 40 students, 25 play cricket, 20 play hockey and 10 play both.
120 people are surveyed. 60 like A, 45 like B and 35 like C. 20 like A and B, 15 like A and C, 10 like B and C, and 5 like all three.
In the class of 40, 25 play cricket and 20 play hockey, with 10 playing both. How many play at least one?
Using the same class of 40, how many play neither game?
What does n(A ∪ B) count?
In Venn terms, "All A are B" asserts that which region is empty?
If A = {1, 2, 4} and B = {1, 3, 4, 5, 7}, how many ordered pairs are in A × B?
Without looking back: what does the subtraction in inclusion-exclusion remove, and how do you count the people who play neither game?
The answer: it removes the double-counted overlap, and "neither" is the total minus the union.
The items in both A and B form the . The formula for two sets is n(A) + n(B) − n(A ∩ B), where the subtracted term removes the double-counted . Everyone in at least one set forms the , and the number in neither is the total minus that union, which for the class of 40 is . If A has 3 elements and B has 5, then A × B has ordered pairs.
In a survey of 60 people, 35 read newspaper P, 25 read newspaper Q, and 15 read both. How many read at least one, how many read exactly one, and how many read neither?
Write the overlap first: 15 people read both. Then P only = 35 − 15 = 20, and Q only = 25 − 15 = 10. At least one is the union, 35 + 25 − 15 = 45, matching 20 + 10 + 15.
Exactly one excludes the overlap, so it is 20 + 10 = 30. Neither is the complement of the union against the total: 60 − 45 = 15.
Check the arithmetic against the whole survey: the four regions are P only 20, Q only 10, both 15 and neither 15, and they sum to 60. If that sum misses the total, one of the regions has been miscounted - which is why the final consistency check is worth the ten seconds.
Sets give you a picture for every "all", "some" and "none" claim, and a formula for every "at least", "exactly" and "neither" count. The next lesson leaves circles for a different kind of spatial reasoning: relations and directions, where a set of statements about who is where is turned into a diagram and measured.