Warning: The intersection values provided are mathematically impossible (result in negative set sizes). Please verify your inputs.
Understanding Venn Diagram Calculations
A Venn Diagram is a powerful visual tool used in set theory, probability, and logic to represent the relationships between different groups of items. This calculator uses the Principle of Inclusion-Exclusion to determine the size (cardinality) of different regions within two or three sets.
The Three-Set Formula
To find the total number of unique elements in three overlapping sets (the Union), we use this formula:
This formula ensures that elements present in multiple sets are not double-counted (or triple-counted).
Practical Example: Student Enrollment
Imagine a school where students can enroll in three different clubs: Art (A), Band (B), and Chess (C).
Art (A): 50 students
Band (B): 40 students
Chess (C): 30 students
Art & Band: 15 students
Art & Chess: 10 students
Band & Chess: 8 students
All Three: 5 students
To find students Only in Art, you subtract the overlaps from the total Art group: 50 – (10 – 5) – (5) – (15 – 5) = 30. Using the formula for the Union, we find there are 92 unique students participating in at least one club.
Key Terms
Cardinality: The number of elements in a set.
Intersection (∩): The elements that belong to two or more sets simultaneously.
Union (∪): The collection of all elements belonging to any of the sets.
Complement: Elements that are NOT in a specific set (often represented as the surrounding "Universal Set").
function calculateVenn() {
// Input Retrieval
var a = parseFloat(document.getElementById("setA").value) || 0;
var b = parseFloat(document.getElementById("setB").value) || 0;
var c = parseFloat(document.getElementById("setC").value) || 0;
var ab = parseFloat(document.getElementById("intAB").value) || 0;
var ac = parseFloat(document.getElementById("intAC").value) || 0;
var bc = parseFloat(document.getElementById("intBC").value) || 0;
var abc = parseFloat(document.getElementById("intABC").value) || 0;
// Calculation logic based on Inclusion-Exclusion Principle
// Components
var onlyABC = abc;
var onlyAB = ab – abc;
var onlyAC = ac – abc;
var onlyBC = bc – abc;
var onlyA = a – (onlyAB + onlyAC + onlyABC);
var onlyB = b – (onlyAB + onlyBC + onlyABC);
var onlyC = c – (onlyAC + onlyBC + onlyABC);
// Total Union
var union = a + b + c – (ab + ac + bc) + abc;
// Output Mapping
document.getElementById("resUnion").innerText = union;
document.getElementById("resOnlyA").innerText = onlyA;
document.getElementById("resOnlyB").innerText = onlyB;
document.getElementById("resOnlyC").innerText = onlyC;
document.getElementById("resOnlyAB").innerText = onlyAB;
document.getElementById("resOnlyAC").innerText = onlyAC;
document.getElementById("resOnlyBC").innerText = onlyBC;
// Show results
document.getElementById("vennResults").style.display = "block";
// Error handling for impossible sets
var errorWarning = document.getElementById("errorWarning");
if (onlyA < 0 || onlyB < 0 || onlyC < 0 || onlyAB < 0 || onlyAC < 0 || onlyBC < 0 || onlyABC < 0) {
errorWarning.style.display = "block";
} else {
errorWarning.style.display = "none";
}
}