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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.

Kugwira ntchito chitsanzo · {1,2,3,4} union {3,4,5,6}

Set-Theory · Sets

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

Gawo ndi Gawo

  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|.

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

Tsopano inu Sankhani vuto, kapena kulemba kapena kujambula wanu. Pakalipano, yankho limeneli lidzapezeka patsamba lino.

Kusunga ntchito yanu yokha

M’malo mwake, maakaunti aulere amawonjezera malemba pazophunzira zonse, mnda wa zomwe mwamaliza, mavuto anu omaliza m’malo limodzi, ndi m’bale amene mungafunse za nkhaniyo. Maphunziro a matekinoloje ndi otsegulira kwa aliyense, olembetsa kapena osalembetsa.

Kulembetsa Kulowa

Zithunzi

Pitani pa dzina lililonse lachifaniziro kuti mudziwe tanthauzo lake, chithunzi chake, ndi zimene limatanthauza.

Momwe mungagwiritsire ntchito: Pigeonhole principle and inclusion, exclusion

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

Mafunso omwe anthu amafunsa

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.

Zambiri pa 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