Inclusion and Exclusion

# Inclusion and Exclusion - Math 100 Counting via...

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Unformatted text preview: Math 100 Counting via Inclusion/Exclusion Martin H. Weissman 1. Inclusion/Exclusion We begin with the following questions: (1) How many even numbers are there between 1 and 100 (inclusive)? (2) How many multiples of three are there between 1 and 100 (inclusive)? (3) How many multiples of six are there between 1 and 100 (inclusive)? (4) How many numbers between 1 and 100 (inclusive) are multiples of 2, but not multiples of six? (5) How many numbers between 1 and 100 (inclusive) are multiples of 2, or multiples of 3, but not both? (6) How many numbers between 1 and 100 (inclusive) are multiples of 2, or multiples of 3, or multiples of 5, but not multiples of 6, nor 15, nor 10? When S is a set, we write # S for its cardinality , i.e., for the number of elements in S . For example: # { 2 , 3 , 5 } = 3 . Let’s rephrase the questions above using set theory. Let E be the set of even numbers between 1 and 100. Let T be the set of multiples of three between 1 and 100. Let F be the set of multiples of 5 between 1 and 100. Notice that E ∩ T is the set of multiples of 6 between 1 and 100. Now the questions can be rephrased: (1) What is # E ? (2) What is # T ? (3) What is #( E ∩ T )? (4) What is # E- #( E ∩ T )? (5) What is #( E ∪ T )- #( E ∩ T )? (6) Challenge for later... There are a few basic theorems to help you with these kind of counting problems. The first is: Theorem 1. Let S and T be two sets. Then, #( S ∪ T ) = # S + # T- #( S ∩ T ) ....
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Inclusion and Exclusion - Math 100 Counting via...

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