Combinatorics

6 the principle of inclusion exclusion can be used to

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Unformatted text preview: n ≤ 300)∧ (3|n)} B = {n∈Z | (1 ≤ n ≤ 300)∧ (5|n)} C = {n∈Z | (1 ≤ n ≤ 300)∧ (7|n)} •  How big are these sets? We use the floor func5on |A| = ⎣ྏ300/3⎦ྏ = 100 |B| = ⎣ྏ300/5⎦ྏ = 60 |C| = ⎣ྏ300/7⎦ྏ = 42 CSCE 235 Combinatorics 12 Applica5on of PIE: Example A (2) •  How many integers between 1 and 300 (inclusive) are divisible by at least one of 3,5,7? Answer: |A∪B ∪C| •  By the principle of inclusion- exclusion |A∪B ∪C|= |A|+|B|+|C|- [|A∩B|+|A∩C|+|B∩C|]+|A∩B∩C| •  How big are these sets? We use the floor func5on |A| = ⎣ྏ300/3⎦ྏ = 100 |A∩B| = ⎣ྏ300/15⎦ྏ = 20 |B| = ⎣ྏ300/5⎦ྏ = 60 |A∩C| = ⎣ྏ300/21⎦ྏ = 100 |C| = ⎣ྏ300/7⎦ྏ = 42 |B∩C| = ⎣ྏ300/35⎦ྏ = 8 |A∩B∩C| = ⎣ྏ300/105⎦ྏ = 2 •  Therefore: |A∪B ∪C| = 100 + 60 + 42 - (20+14+8) + 2...
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This note was uploaded on 10/31/2013 for the course CSCE 235 taught by Professor Staff during the Spring '08 term at UNL.

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