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Solutions_3_W_Merged - MAT 312/AMS 351 Summer 2011...

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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 X containing a and 2 n subsets of X not containing a . In total, we have 2 n + 2 n = 2 n +1 subsets of X . 4. (10 points) See picture below.
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