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Pigeonhole principle and inclusion, exclusion

Two counting ideas that prove things without listing anything.

Pigeonhole: more pigeons than holes means some hole holds two. Inclusion, exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|, and the alternating sum continues for more sets. Picture it: a Venn diagram; the overlap is what you counted twice. Think it: inclusion, exclusion is the Möbius function of the subset lattice.

Nhazi ahụ ejirila · {1,2,3,4} union {3,4,5,6}

Set-Theory · Sets

{1,2,3,4} union {3,4,5,6}

Nzọụkwụ site n'ụdị

  1. A = \left\{1, 2, 3, 4\right\},\quad B = \left\{3, 4, 5, 6\right\}

    The two sets.

  2. A \cup B = \left\{1, 2, 3, 4, 5, 6\right\}

    Keep everything in either set.

  3. |A| = 4,\ |B| = 4,\ |A \cup B| = 6

    Cardinalities. Note |A ∪ B| = |A| + |B| − |A ∩ B|.

Gosi nzaghachi
A \cup B = \left\{1, 2, 3, 4, 5, 6\right\}

Ugbu a ị Họrọ nsogbu, mọọbụ tinye mọọbụ depụta gị. Nzọụkwụ ọbụla, inyogo, azịza ahụ echekwara ruo mgbe ịjụla.

Kwụsị ọrụ gị

Akwụkwọ ndebanye n'efu na-egbakwunye ntọala n'ụdị ọmụmụ ihe ọbụla, ndepụta nke ihe ị ga-eme, nsogbu ị ga-ewepụ n'otu ebe, na onye nkụzi ị ga-ajụ banyere ihuakwụkwọ a. Akwụkwọ ndepụta ihe bụ nke ọbụla mepere maka onye ọbụla, ọbụla ma ọ bụ na ọ bụghị.

Akaụntụ Nbanye

Aịkọn ndị ahụ ejirila ebe a

Tinye ọbụla simbụl maka nkọwa zuru ezu, inyogo, na ihe ọbụla ọbụla na-egosi.

Olee otú: Pigeonhole principle and inclusion, exclusion

  1. The two sets.
  2. Keep everything in either set.
  3. Cardinalities. Note |A ∪ B| = |A| + |B| − |A ∩ B|.

Ajụjụ ndị mmadụ jụrụ

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

Oge Combinatorics & Graph Theory

The counting principlesBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs