Venn Diagram & Set Calculator
Set operations with the regions shaded, and Venn word problems solved region by region.
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About the Venn Diagram & Set Calculator
Two calculators in one. Set operations: list the elements of up to three sets A, B and C and a universal set U — numbers, words, ranges such as 1..20, ordered pairs or sets inside sets — and type any expression: (A ∪ B)′ ∩ C, A − B, A Δ B, P(A), A × B, n(A ∪ B), or a statement such as A ∩ B ⊆ C or 3 ∈ A. The calculator shows how the expression was read, works out every part of it step by step, lists the result and shades its regions in the Venn diagram, with the elements of every region.
Word problems: for “60 people were asked which newspapers they read…” problems, type what you know — totals, each set, the overlaps, “only”, “neither”, “exactly two”, as counts or as probabilities with conditional probabilities such as P(A | B). The calculator turns every fact into an equation in the regions of the diagram and solves them exactly, so it can work backwards from any combination of facts. It fills in every region, answers the usual questions (at least one, exactly one, none), names the fact that contradicts the others, and when the facts leave a region open it gives the smallest and largest value that region can take.
How to use it
- Choose Set operations to combine sets you can list, or Word problem to solve a problem given as numbers.
- For set operations, type the elements of A (and B, C and U if you need them) separated by commas, then the expression. The symbol buttons insert ∪ ∩ − Δ ′ × and the rest; typed words such as union, intersection and minus work too. Complements need the universal set U.
- For a word problem, choose counts or probabilities and two or three sets, name the sets, and fill in only the numbers you know. Put anything else in More facts, one per line, such as exactly two = 12 or n(A ∩ B′) = 7.
- Read the result, the shaded diagram and the region table, then the working; tap a region of the diagram to see what is in it.
- Copy the result with its working, or download a CSV file or the diagram as PNG or SVG — with a Pro pass or after unlocking this result; without one you see the free preview.
Examples
U = {1, 2, …, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}, C = {3, 4, 5, 6}; (A ∪ B)′ and (B − C)′(A ∪ B)′ = {5, 7, 9}; (B − C)′ = {1, 3, 4, 5, 6, 7, 9}U = 1..10, A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7}; (A ∪ B)′ = A′ ∩ B′true: both sides are {8, 9, 10}60 people: 25 read H, 26 read T, 26 read I; 11 read H and T, 9 H and I, 8 T and I; 3 read all three
52 read at least one paper, 30 exactly one, 19 exactly two and 8 none
By inclusion–exclusion, n(H ∪ T ∪ I) = 25 + 26 + 26 − 11 − 9 − 8 + 3 = 52.
Of 400 people, 250 speak Hindi and 200 English, and everyone speaks at least one of them
50 speak both, 200 only Hindi and 150 only English
Of 50 people, 30 drink tea and 25 coffee
Both: between 5 and 25; neither: between 0 and 20
Without one more fact (how many drink both, or neither), the overlap is not fixed.
P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.2
P(A ∪ B) = 0.7, P(neither) = 0.3, P(A | B) = 0.5; A and B are independent
Common uses
- Checking homework on unions, intersections, complements, power sets and Cartesian products, with the working.
- Shading the region of an expression such as (A ∩ B)′ ∪ C to see what it means, or to check a drawing.
- Solving survey problems: how many people chose exactly one option, two options, or none.
- Finding probabilities of combined events, and testing whether two events are independent.
- Comparing lists of names, codes or products to find what they share and what is unique to each.
The operations and how an expression is read
- Union A ∪ B: elements in A or B (or both). Intersection A ∩ B: elements in both.
- Difference A − B: in A but not in B. Symmetric difference A Δ B: in exactly one of them.
- Complement A′: the elements of the universal set U that are not in A.
- Power set P(A): the set of all 2ⁿ subsets of a set with n elements. Cartesian product A × B: every ordered pair (a, b).
- Relations: A ⊆ B (every element of A is in B), A ⊂ B (a subset and not equal), A = B, and x ∈ A.
Complements are worked out first, then ×, then ∩, then ∪, − and Δ from left to right, so A ∪ B ∩ C means A ∪ (B ∩ C). The calculator always writes the expression back with brackets so you can see how it was read. The definitions are those of NCERT Class 11 Mathematics, chapter 1.
Inclusion–exclusion
For two sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B): the overlap is counted twice in the sum, so it is taken off once. For three sets, n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C), the inclusion–exclusion principle. The single regions follow the same way: “A only” = n(A) − n(A ∩ B) − n(A ∩ C) + n(A ∩ B ∩ C), and “A and B only” = n(A ∩ B) − n(A ∩ B ∩ C).
How word problems are solved
A diagram of two sets has 4 regions and one of three sets has 8 (counting the region outside every circle). Each fact you give is an equation in those regions: “n(A ∩ B) = 11” says that the two regions inside both A and B add up to 11. The calculator solves the equations exactly, so it does not matter which facts you know, as long as there are enough of them. Facts that cannot all be true are reported, with the one that does not fit. When a region is not fixed, the calculator finds its smallest and largest possible values, keeping every region at 0 or more (two small linear programs per region); counts are whole numbers, so these bounds are rounded inwards.
Probabilities
With probabilities, P(U) = 1 and every region is between 0 and 1. A conditional probability is a ratio, P(A | B) = P(A ∩ B) / P(B), which the calculator turns into the equation P(A ∩ B) − P(A | B)·P(B) = 0. Two events are independent when P(A ∩ B) = P(A)·P(B), and mutually exclusive when P(A ∩ B) = 0; once every region is known, the calculator checks both for every pair, as defined in NCERT Class 12 Mathematics, the chapter on probability.
Limitations
- Up to three sets A, B and C and a universal set U; each set can have up to 10,000 elements.
- Power sets are listed for sets of up to 12 elements (4,096 subsets) and Cartesian products up to 10,000 pairs.
- Word problems take two or three sets; facts must be sums of regions (or conditional probabilities), so a fact such as “twice as many like tea as coffee” goes in as an equation of regions only if you rewrite it as one.
- Elements are compared exactly: 1, 1.0 and 2/2 are the same number, but “Apple” and “apple” differ unless you tick Ignore upper and lower case.
Privacy
On this device: nothing is sent
Everything happens in your browser. What you enter or open here is not uploaded or stored by MySmartCoPilot.
Frequently asked questions
What do I get without a pass?
Without a pass, Venn Diagram & Set Calculator shows your sets as entered and a watermarked diagram of the circles, with the result, the shaded regions, every region count and the working hidden. Until you unlock it, the result can’t be downloaded or copied. A Pro, Premium or Ultimate pass, a one-time payment that never renews, unlocks the full result. The pricing page lists the passes and their prices.
How do I find the complement of a set?
Fill in the universal set U — everything the sets are taken from — and type A′ (or A'). With U = {1, …, 9} and A = {1, 2, 3, 4}, A′ = {5, 6, 7, 8, 9}. Without U a complement cannot be worked out, because it depends on what is outside A.
How do I shade (A ∪ B)′ in a Venn diagram?
Type the expression and the calculator shades its regions: (A ∪ B)′ is the part of U outside both circles. It also lists the elements of each region, so you can check which elements end up in the shaded part.
How do I solve a Venn diagram word problem with three sets?
Start from the middle: the number in all three, then each “two only” region (the pair’s total minus the middle), then each “one only” region, and finally “none” from the total. Choose Word problem, Three sets and type the numbers you know; the calculator does these steps and fills in every region.
What is the difference between A − B and A Δ B?
A − B keeps what is in A and not in B. A Δ B keeps what is in exactly one of the two sets, so it equals (A − B) ∪ (B − A). For A = {1, 2, 3} and B = {3, 4}, A − B = {1, 2} and A Δ B = {1, 2, 4}.
How many subsets does a set have?
A set with n elements has 2ⁿ subsets, counting the empty set and the set itself, and 2ⁿ − 1 proper subsets. {a, b, c} has 8 subsets; type P(A) to list them.
Why does the calculator give a range instead of a number?
Because the facts given are not enough to fix that region: several diagrams fit all of them. The range shows every value that is still possible. One more fact, such as the size of an overlap or of “neither”, usually fixes it.