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Unformatted text preview: MAT 312/AMS 351, Summer 2011 Solutions to Homework Assignment 3 Maximal grade for HW3: 100 points Section 2.1 2. (10 points) List all the subsets of the set X = { a,b,c } . How many are there? Next, try with X = { a,b,c,d } . Now suppose that the set X has n elements: how many subsets does X have? Try to justify your answer. Answer: 2 n . Solution: For X = { a,b,c } we have the following list of 2 3 = 8 subsets: ∅ , { a } , { b } , { c } , { a,b } , { a,c } , { b,c } , { a,b,c } For X = { a,b,c,d } we have the following list of 2 4 = 16 subsets: ∅ , { a } , { b } , { c } , { d } , { a,b } , { a,c } , { a,d } , { b,c } , { b,d } , { c,d } , { a,b,c } , { a,b,d } , { a,c,d } , { b,c,d } , { a,b,c,d } In general, if X has n elements, then there are 2 n subsets of X . Let us prove it using induction by n . Base case: n = 1, there are two subsets X and ∅ . Step: Suppose that X has n +1 elements and a ∈ X . By the assumption, there are 2 n subsets of X \ { a } , hence there are 2 n subsets of...
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This note was uploaded on 01/25/2012 for the course MAT 312 taught by Professor Bescher during the Summer '08 term at SUNY Stony Brook.
 Summer '08
 BESCHER
 Algebra, Sets

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