Quick drill with answers

Quick drill with answers - { a,c } . 6. Suppose that S is a...

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Math 100 Answers to Drill Questions Martin H. Weissman 1. List all of the subsets of { a,b } The subsets of { a,b } are: , { a } , { b } , { a,b } . 2. How many subsets does { a,b,c } have? There are 8 subsets of { a,b,c } , since { a,b,c } has 3 elements, and 8 = 2 3 . 3. How many whole numbers are there between 6 and 100, inclusive? The whole numbers between 6 and 100 are in bijection with the whole num- bers between 1 and 95. Hence there are 95 whole numbers between 6 and 100. 4. How many multiples of 3 are there between 6 and 100, inclusive? The multiples of 3 between 6 and 100 can be listed: 6 , 9 , 12 ,..., 99. These are in bijection with the numbers 2 , 3 , 4 ,..., 33 (by dividing the elements of the previous list by 3). These are in bijection with the numbers 1 , 2 , 3 ,..., 32. Hence, there are 32 multiples of 3 between 6 and 100. 5. How many subsets of { a,b,c } have two elements? There are 3 subsets of { a,b,c } with two elements. They can be directly counted: { a,b } , { b,c } , and
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Unformatted text preview: { a,c } . 6. Suppose that S is a set with 6 elements. How many subsets does S have with 3 elements? There are 20 subsets of S with 3 elements, since (6!) / (3!3!) = 20. 7. Suppose that S is a set with 5 n elements. How many subsets of S have 2 n elements? There are (5 n )! (2 n )!(3 n )! subsets of S with 2 n elements. 8. Suppose that S is a set with 6 elements. How many subsets of S have an even number of elements? There are 32 subsets of S with an even number of elements, since 32 = 2 6-1 . 9. Suppose that S is a set with 10 elements. How many ordered pairs are there, whose entries are elements of S ? There are 100 ordered pairs with entries in S , since 100 = 10 2 . 10. Let S be the set of pairs ( a,b ), such that 1 a 100, 1 b 100, and a + b = 101. How many elements does S have? S has 100 elements....
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