Venn Diagrams You Can Shade: Sets, Operations and Counting

How to use this simulation

1. In “Operations and shading” pick one operation at a time (A ∪ B, A ∩ B, A − B, A′ …). The matching region fills in; before each click, predict which regions will light up.
2. Switch to “Three sets”: the picture now has eight regions. Compare A ∩ B ∩ C, (A ∩ B) − C and A − (B ∪ C) and note where each sits.
3. Open “Place the numbers”: the integers 1 to 12 glide into their regions. Change A and B (even, multiples of 3, primes, factors of 12 …) and check the roster of the answer in the readings.
4. In “Counting formula” slide n(A), n(B) and n(A ∩ B). The circle areas and the overlap are drawn to scale and n(A ∪ B) = n(A) + n(B) − n(A ∩ B) is evaluated live. Impossible values are corrected and flagged.
5. In “De Morgan’s laws” switch the right-hand picture on and off. Both sides shade the same region, so the two sides of the law are equal.

Cricket, football and a class of forty

A class has 40 students. 25 play cricket, 18 play football and 8 play both. How many play at least one of the two? Many people add 25 and 18 and announce 43. But 43 is more than the 40 students in the room. Something went wrong.

What went wrong is that the 8 students who play both were counted twice, once in the cricket group and once in the football group. A student is still one student. So subtract the double-counted group once: 43 − 8 = 35. Exactly 35 play at least one game, and the other 5 play neither.

The easiest way to avoid double counting is a picture: a big rectangle for the whole class and two circles that overlap a little. The overlap is the “plays both” group. That picture is a Venn diagram, and the simulation above draws it for you, with regions you can shade and numbers you can count yourself.

Once the picture exists, the arithmetic is almost free: cricket only 17, football only 10, both 8, neither 5. The four regions add up to exactly 40. Those four regions are the heart of the whole chapter.

Starting from zero: what a set is

A set is a well-defined collection of objects. “Well-defined” matters: for any object you must be able to say for sure whether it is in the set. “The students in your class with an even roll number” is a set, because each roll number settles it. “The nicest students in your class” is not, because nobody agrees on nice.

Each object in a set is called an element or member. The number 4 is an element of the set of even numbers, written 4 ∈ A. The number 5 is not, so 5 ∉ A. Sets are usually named with capital letters (A, B, C). Order and repeats do not change a set: {1, 2, 3} and {3, 1, 2, 2} are the same set.

The universal set U is the big set from which all the sets in a discussion are drawn. On this page U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. It depends on the context: for a talk about natural numbers U might be all natural numbers, and in the cricket story U is the whole class. In a Venn diagram U is the rectangle, and each set is a circle inside it.

Learning to read the regions is the real skill. Two sets give four regions: only A, only B, both (the overlap) and neither (outside both circles but inside the rectangle). Three sets give eight regions. Every set operation is simply a rule for which of these regions to shade.

Key words side by side

Definition questions come from this vocabulary. Match each symbol to its picture rather than memorising in the dark.

Two ways to write a set: roster and set-builder

Roster form lists the elements between curly brackets: A = {2, 4, 6, 8, 10, 12}. Set-builder form states the rule instead: A = {x ∈ U : x is even}, read “A is the set of all x in U such that x is even”. The two forms describe the same set.

When is each better? Roster form is clear when there are few elements. Set-builder form wins when the set is huge (all even numbers) or when the rule is the whole point. In the simulation you choose the rule (set-builder form) and the readings panel shows the roster form for you.

A few more rules and their rosters: the factors of 12 give {1, 2, 3, 4, 6, 12} (6 elements), the perfect squares give {1, 4, 9}, and the odd numbers give {1, 3, 5, 7, 9, 11}. Changing how a set is written never changes the set, only how easy it is to read.

Subsets, the empty set and the power set

If every element of A is also in B, then A is a subset of B, written A ⊂ B. For example {4, 8, 12} is a subset of the even numbers {2, 4, 6, 8, 10, 12}. Every set is a subset of itself, and the empty set ∅ is a subset of every set. A subset that is not the whole set is a proper subset.

The power set P(A) is the set of all subsets of A. For {a, b, c} the subsets are ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}: 8 in all. The rule is simple. Each element is either in a subset or out of it, which is two choices per element. So a set with n elements has 2 × 2 × … (n times) = 2ⁿ subsets. A 4-element set has 16, a 5-element set 32 and a 10-element set 1,024.

Every operation and its region

Now the main work. In the examples below U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, A = {2, 4, 6, 8, 10, 12} (even numbers) and B = {3, 6, 9, 12} (multiples of 3). The “Place the numbers” mode starts with exactly these sets, so you can compare the page with the picture.

Three sets in one picture

Three circles make eight regions. In the simulation switch to “Three sets” and take C = prime numbers = {2, 3, 5, 7, 11}. Then A ∩ B ∩ C = ∅: no number is even, a multiple of 3 and prime all at once. That is a nice example of the empty set; the middle region of the picture stays blank.

A ∪ B ∪ C = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} (11 elements, everything except 1). (A ∩ B) − C = {6, 12}, because 6 and 12 are even multiples of 3 but not prime. A − (B ∪ C) = {4, 8, 10}, the even numbers that are neither multiples of 3 nor prime. And A ∩ (B ∪ C) = {2, 6, 12}.

Look at the pairwise overlaps too: A ∩ C = {2}, B ∩ C = {3} and the triple overlap is ∅. Those numbers feed the three-set counting formula later.

De Morgan’s laws

De Morgan’s laws say that taking a complement swaps “and” with “or”. In words: the complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements.

Counting with Venn diagrams: the union formula

Counting elements of two sets counts the overlap twice, so it must be removed once. That is the whole formula, and it is the cricket story in symbols.

Experiments to try in the simulation

Before each experiment make a prediction, then check it against the picture.

• In “Counting formula” the starting values are n(U) = 30, n(A) = 14, n(B) = 11 and n(A ∩ B) = 5. Work out n(A ∪ B) yourself and then read 20 in the readings. 10 elements sit outside both circles.
• Slide n(A ∩ B) down to its minimum. The circles slide apart until they are separate, and n(A ∪ B) = n(A) + n(B).
• Slide n(A ∩ B) up to its maximum, the smaller of n(A) and n(B). The smaller circle sinks inside the bigger one: B ⊂ A.
• Keep lowering n(U) until n(A) + n(B) is more than n(U). The simulation raises the minimum allowed overlap and the status line warns you, because A ∪ B can never be bigger than U.
• In “Operations and shading” alternate between A − B and B − A. The two regions differ, proving that difference depends on order.
• In “Place the numbers” set A to even numbers and B to odd numbers. The overlap is empty because no number is both, and A ∪ B is the whole universal set.
• In De Morgan mode toggle the right-hand picture on and off many times and notice that the two sides never get a chance to differ.

Solved problems

Every number below is computed by the same code that drives the simulation, so the page and the picture agree.

Common mistakes

These slips cost the most marks in exams.

• Adding n(A) and n(B) and calling it n(A ∪ B) without removing the overlap. Remember the cricket and football story.
• Treating A − B and B − A as the same. They are usually different crescents in the picture.
• Taking a complement without knowing U. A′ means “what is in U but not in A”, which makes no sense without U.
• Mixing up ∈ and ⊂. 4 ∈ A (element), but {4} ⊂ A (subset). An object on the left takes ∈; a set on the left takes ⊂.
• Treating ∅, {∅} and {0} as one thing. ∅ has no elements; {∅} has one (the empty set); {0} has one (the number zero).
• In De Morgan’s laws, adding the prime mark but forgetting to switch the symbol. (A ∪ B)′ = A′ ∪ B′ is wrong: the ∪ must become ∩.
• Reading “or” as “one but not both”. In sets, ∪ includes the overlap.

Where Venn diagrams show up in real life

Type “tea AND coffee” in a search box and the engine intersects; “tea OR coffee” unions; “tea NOT coffee” takes a difference. In databases the SQL JOIN and UNION commands are the computer versions of exactly these operations.

Filters on a shopping site do the same: “brand A or B” together with “price under 500” is a union followed by an intersection. Surveys, election statistics, matching symptoms to diseases, subject lists in a library: it is the same picture each time.

In probability, events are sets: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) is this page’s counting formula in probability form. In digital circuits the AND, OR and NOT gates are intersection, union and complement.

Exam corner: Grade 7–11 and board exams

Boards such as NCTB (Bangladesh SSC), CBSE and ICSE test sets with definitions, drawing a Venn diagram, verifying an identity such as De Morgan’s laws, and word problems that need n(A ∪ B). Verification questions expect the left side and the right side to be worked out separately and then compared.

Revision: the last-minute summary

The night before an exam, this list and the vocabulary table above are all you need to skim.

• A set is a well-defined collection; elements use ∈ and ∉; the universal set is U.
• Roster form lists the elements; set-builder form states the rule.
• A ∪ B is in at least one; A ∩ B is in both; A − B is in A but not B; A′ is U − A.
• A △ B = (A − B) ∪ (B − A) is in exactly one.
• De Morgan: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
• n(A ∪ B) = n(A) + n(B) − n(A ∩ B); with three sets add n(A ∩ B ∩ C) at the end.
• A set with n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets.
• Draw the picture, write a number in each region, then add and subtract.

Frequently asked questions