Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Discrete Mathematics → Combinatorics ↓


Inclusion-Exclusion Principle


The inclusion-exclusion principle is an essential concept in combinatorics, a branch of discrete mathematics that deals with counting and enumerating possible outcomes. This principle helps us determine the size of the union of multiple sets when the sizes of the individual sets and their intersection are known. This principle is not only powerful but is widely applicable in various fields ranging from probability and statistics to computer science and logic.

Original idea

Let's start with the simplest case: two sets, A and B. The theorem states that to find the number of elements in the union of two sets, A and B (denoted as |A ∪ B|), we should start by adding the number of elements in each set separately. However, when we do this, we count the elements that are common to both sets, A ∩ B. Therefore, we should subtract the number of elements in the intersection of the two sets to correct this double counting.

In set theory the formula for the combination of two sets is described as follows:

|A ∪ B| = |A| + |B| - |A ∩ B|

Visualise the concept

Consider an example where we have two overlapping circles or groups:

A B

In this illustration, the blue circle represents the set A, the green circle represents the set B, and the region where they overlap is the intersection A ∩ B.

Expansion to three sets

When dealing with three sets, such as A, B and C, the theory gets a little more complicated. For the union of three sets, the inclusion-exclusion principle tells us the following:

|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

Here, we first add the sizes of the individual sets. Then we subtract the sizes of all pairwise intersections to correct for double counting. However, this subtraction also discards elements that are the same three times (for each pairwise subtraction) in all three sets, so we add the size of the triple intersection back once.

Step-by-step example with three sets

Let us consider a practical example:

  • Set A represents people who like apples.
  • Set B represents people who like bananas.
  • Set C represents people who like cherries.

Suppose we have the following data:

  • |A| = 30 (people like apples)
  • |B| = 25 (people like bananas)
  • |C| = 20 (people like cherries)
  • |A ∩ B| = 10 (same as both apples and bananas)
  • |A ∩ C| = 5 (same as both apple and cherry)
  • |B ∩ C| = 8 (same as both banana and cherry)
  • |A ∩ B ∩ C| = 3 (all three same as fruits)

Applying the Inclusion-Exclusion Principle:

|A ∪ B ∪ C| = 30 + 25 + 20 – 10 – 5 – 8 + 3
            = 60

Thus, 60 people like at least one of the three types of fruits.

Generalization to n sets

The inclusion-exclusion principle can be generalized to more than three sets. Suppose we have n sets, A1 , A2 , ..., An. The principle is generalized as follows:

|A1 ∪ A2 ∪ ... ∪ An | ,
∑ |Ai | - ∑ |Ai ∩ Aj | + ∑ |Ai ∩ Aj ∩ Ak | - ... + (-1)n+1 |A1 ∩ A2 ... ∩ An |

Here, the sums are taken based on increasing number of intersections, and the sign changes between subtraction and addition depending on the number of sets in each intersection.

Visual example for three sets

Here's a more detailed illustration of the three overlapping sets:

A B C

The three circles represent sets. The three paired intersections and the center intersection represent regions subtracted and then added back according to the inclusion-exclusion formula.

Applications and advanced examples

The inclusion-exclusion principle is incredibly versatile and can be applied in many scenarios, such as:

1. Counting restricted system

Suppose we are asked to count the number of permutations of the numbers {1, 2, ..., n} such that none of the numbers appear in their original positions (a perturbation). The inclusion-exclusion principle allows us to systematically include and exclude those calculations where a certain number of items are in their original positions.

2. Estimating the probability

In probability, the principle can be used to calculate the probability of the union of multiple events. If we know the probability of individual events and their intersection, we can use the inclusion-exclusion principle to find the probability of at least one of the events occurring.

3. Network reliability

In network design, this principle can assess the reliability of complex systems by calculating the probability that at least one critical component will fail.

Working on a more complex example

Let's look at a more complex example of four sets:

Consider four groups of students:

  • Set A: Students playing football
  • Set B: Students playing basketball
  • Set C: Students playing cricket
  • Set D: Students playing tennis

Assume we have the following data:

  • |A| = 40
  • |B| = 35
  • |C| = 25
  • |D| = 20
  • |A ∩ B| = 15
  • |A ∩ C| = 10
  • |A ∩ D| = 5
  • |B ∩ C| = 10
  • |B ∩ D| = 8
  • |C ∩ D| = 6
  • |A ∩ B ∩ C| = 4
  • |A ∩ B ∩ D| = 2
  • |A ∩ C ∩ D| = 1
  • |B ∩ C ∩ D| = 3
  • |A ∩ B ∩ C ∩ D| = 1

Using the Inclusion-Exclusion Principle:

|A ∪ B ∪ C ∪ D| = |A| + |B| + |C| + |D|
                    - (|A ∩ B| + |A ∩ C| + |A ∩ D| + |B ∩ C| + |B ∩ D| + |C ∩ D|)
                    + (|A ∩ B ∩ C| + |A ∩ B ∩ D| + |A ∩ C ∩ D| + |B ∩ C ∩ D|)
                    − |A ∩ B ∩ C ∩ D|
                 = 40 + 35 + 25 + 20
                   - (15 + 10 + 5 + 10 + 8 + 6)
                   + (4 + 2 + 1 + 3)
                   - 1
                 = 120 – 54 + 10 – 1
                 = 75

Hence, 75 students play at least one of the four sports.

Special considerations and tips

  • When using this principle, make sure you calculate the intersections correctly. Ignoring or miscalculating them can lead to incorrect results.
  • This method involves alternating sums and differences, so it is important to keep track of the signals for accuracy.
  • Complexity increases with more sets, so careful data organization is important to avoid errors.

Conclusion

The inclusion-exclusion principle is an important tool in combinatorics that allows for accurate calculations and probability computations in complex scenarios. Through careful application and consideration of overlapping sets, the principle makes addressing many problems in mathematics and related fields more manageable. By practicing with diverse and increasingly complex examples, one can better understand the mechanics of the principle and its wide range of applications.


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