# 9 - = 3 | ∅ | = |N| = undeFned for now Fact Let A and B...

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Proof A simple counter-example to part 1 is A = { a } ,B = { b } ,C = { c } , where a,b,c are different objects. In this example, B C = and A ( B C ) = { a } whereas A B = and ( A B ) C = { c } . A simple counter- example to part 2 is A = { a } ,B = C = { b } , where again a,b are different. ± Question: Under what conditions are A ( B C ) and ( A B ) C equal? Answer: It is simple to see the solution by drawing the Venn diagrams. The sets are equal when A - ( B C ) = and C - ( A B ) = . 2.2.3 Size of Finite Sets In this section, we begin to explore the number of elements in a fnite set. In section 4.6, we will learn how to talk about the number of elements in an infnite set. D EFINITION 2.8 (C ARDINALITY ) Let A be a Fnite set. The cardinality of A , written | A | , is the number of distinct elements contained in A . Notice the similarity between this deFnition and the length function over lists in Haskell. Here are some examples: |{ a,e,i,o,u }| = 5 |{ a,a,b,c }|
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Unformatted text preview: = 3 | ∅ | = |N| = undeFned for now Fact Let A and B be Fnite sets. Then | A ∪ B | = | A | + | B |-| A ∩ B | . Informal proof The number | A | + | B | counts the elements of A ∩ B twice, so we subtract A ∩ B to obtain the result. A consequence of this proposition is that, if A and B are disjoint sets, then | A ∪ B | = | A | + | B | . 2.2.4 Introducing Power Sets In deFnition 2.2, we deFned a notion of a subset of a set. The set of all subsets of A is called the power set 3 of A . D EFINITION 2.9 (P OWER SET ) Let A be any set. Then the power set of A , written P ( A ) , is { X : X ⊆ A } 3 The name might be due to the cardinality result in proposition 2.10. At least this expla-nation helps to remember the result! 10...
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