2 - LECTURE 2: PARTITIONS AND PRODUCTS ( 1.5-1.6) 1. What...

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Unformatted text preview: LECTURE 2: PARTITIONS AND PRODUCTS ( 1.5-1.6) 1. What is a partition? Consider the following diagram: 1 2 3 4 5 6 7 8 S Weve broken S into pieces that do not overlap, when put together give us S , and none of the pieces are empty. This is the idea behind partitions. To make this more formal we need some definitions. Definition 1. Recall that two sets A and B are disjoint if A B = . We say that a collection of sets is pairwise disjoint if every pair of set from the collection is disjoint. Example 2. Let A 1 = { 3 , 5 , 7 } , A 2 = { 1 , 2 , 4 } , A 3 = { 6 } and A 4 = { 2 } . The collection { A 1 ,A 2 ,A 3 } is pairwise disjoint. But the collection { A 1 ,A 2 ,A 3 ,A 4 } is not pairwise disjoint because A 2 and A 4 are not disjoint. So what is a partition? Definition 3. Fix a set A . A partition of A is a collection (which we will call S ) of subsets of A , which are non-empty, pairwise disjoint, and the union of these sets is all of A . In other words, a partition is a set...
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2 - LECTURE 2: PARTITIONS AND PRODUCTS ( 1.5-1.6) 1. What...

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