2.3 Sets, Venn Diagrams, and Cardinality
Key Takeaways
- Set notation depends on a defined universal set; complements and neither regions cannot be counted without knowing the total population.
- Union means in A or B or both, while intersection means in both A and B; the overlap must be handled once, not twice.
- For two sets, n(A union B) = n(A) + n(B) - n(A intersect B); for three sets, fill the centre overlap first and work outward.
- The Math 30-2 formula sheet includes complement, empty set, intersection, subset, and union notation, so symbol fluency saves time on both logic and probability items.
Sets as Organized Logic
A set is a collection of objects, called elements. In Math 30-2, sets usually describe people in categories, outcomes in a sample space, cards with features, or numbers with properties. The official formula sheet for Mathematics 30-2 lists core logical-reasoning symbols: complement, empty set, intersection, subset, and union. You do not need a long symbolic proof system, but you do need to translate between words, symbols, diagrams, and counts.
The universal set is the full group being considered. If a question surveys 80 students, the universal set is those 80 students, not every student in Alberta. A complement such as A' means everything in the universal set that is not in A. Without the universal set, a complement has no count.
Core notation
| Notation or phrase | Meaning | Venn region |
|---|---|---|
| A union B | In A, in B, or in both | All shaded parts of A and B |
| A intersect B | In both A and B | The overlap |
| A' | Not in A | Every universal-set region outside A |
| A is a subset of B | Every element of A is in B | A sits inside B |
| Empty set | No elements | A region with count 0 |
The word or is inclusive in this setting. If 30 students play soccer or basketball, students who play both are included in that 30. The word and points to the overlap.
Two-set cardinality
Cardinality means the number of elements in a set. For two sets, the central rule is:
n(A union B) = n(A) + n(B) - n(A intersect B)
The subtraction is needed because the overlap was included once in n(A) and once in n(B). If you add both set totals without subtracting the overlap, the students or outcomes in both sets get counted twice.
Worked example: In a group of 50 students, 28 study probability, 22 study functions, and 9 study both. How many study at least one of those topics?
n(P union F) = 28 + 22 - 9 = 41. So 41 students study probability or functions or both. The number studying neither is 50 - 41 = 9. A complete Venn check gives probability only 19, both 9, functions only 13, neither 9; these four regions add to 50.
Three-set diagrams
The bulletin's commentary says students are generally stronger with two sets in a universal set and have more difficulty organizing and interpreting information involving three sets. The fix is a strict fill order.
- Put the triple intersection in the centre first.
- Fill each exactly two overlap by subtracting the centre if the given number includes all three.
- Fill each only-one region by subtracting the relevant overlaps from the set total.
- Add all regions and subtract from the universal total to find neither.
Example: In a class of 36, 18 students chose logic, 16 chose probability, and 14 chose functions for review. Six chose both logic and probability, 5 chose both probability and functions, 4 chose both logic and functions, and 2 chose all three. If pair counts include the students in all three, then the exact two-set overlaps are 4, 3, and 2 after subtracting the centre. The logic-only region is 18 - 4 - 2 - 2 = 10. Probability only is 16 - 4 - 3 - 2 = 7. Functions only is 14 - 2 - 3 - 2 = 7. Total in at least one set is 10 + 7 + 7 + 4 + 3 + 2 + 2 = 35, so 1 student chose none.
Subsets and complements
If A is a subset of B, every element in A is also in B. That does not mean A and B are equal unless both directions are true. For example, the set of students who chose all three topics is a subset of the set of students who chose logic, but the logic set is larger.
Complements often appear as not, did not, outside, neither, or not both. Be careful with not both. It means the complement of the overlap, so it includes only A, only B, and neither. It does not mean neither. The phrase neither A nor B means outside both sets, which is the complement of A union B.
Diploma traps and checks
The fastest way to prevent set errors is to label regions before calculating. Do not write 28 inside the probability-only part if 28 is the total probability count. Totals belong to whole circles; region counts belong inside separated Venn regions.
After filling a diagram, perform two checks:
- Region check: every separated region has one count, even if that count is 0.
- Total check: all regions, including neither, add to the universal-set total.
If a question asks for a probability after the diagram is built, use the requested set as the numerator and the correct sample space as the denominator. For example, "probability a randomly chosen student chose logic or probability" uses n(L union P) over the class total. "Probability a student chose logic given that the student chose probability" uses only the probability circle as the denominator. This is why set fluency supports the Probability domain as well as the Logical Reasoning domain.
In a group of 60 students, 34 take art, 28 take drama, and 12 take both. How many take art or drama or both?
A survey has 45 students. If 31 students are in set M union N, how many students are in neither M nor N?
Which phrase best describes A intersect B?