Alphabeta Math

Inclusion–Exclusion, the Pigeonhole Principle and Double Counting: Examples and Counterexamples

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✓ 10 results · all verified · 1 also independently AI-judged
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1 · Prerequisites
Combinatorics
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

The sieve run in full on three explicit finite sets and then on four, with every nonempty intersection listed

Example

The surjections from a five-element set onto a three-element set counted by the sieve formula and by direct subtraction

Example

All nine derangements of a four-element set listed, and the count checked against the formula and both recurrences

Example

The ratio ι(Dₙ)/ι(n!) computed for small n as a quotient of two counts, with no probability space claimed

Example

In a finite set with a symmetric irreflexive relation and at least two elements, two elements have equally many neighbours

Example

For a finite symmetric irreflexive relation the sum of the neighbour counts is twice the number of unordered related pairs

Example

Distributing a finite set over a finite set of boxes, with the ceiling bound computed and attained

Counterexample

A three-set count that drops the triple intersection and returns the wrong answer

Counterexample

A list of six distinct reals with no strictly increasing sublist of length four and no strictly decreasing sublist of length three

Counterexample

A relation whose row fibres all differ from the average size, so the averaging principle gives a bound that no fibre meets exactly

Sources