Challenging Mathematics to Raise Mathematical Maturity I like to define mathematical maturity as “one’s ability to analyze, understand, and communicate mathematics.” Students with higher levels of mathematical maturity are more likely to perform well on standardized math tests. For more information about mathematical maturity, click on the following link: Mathematical Maturity Challenging Mathematics Today I would like to challenge you a bit while helping you take some steps toward increasing your level of mathematical maturity by presenting you with some more difficult mathematics. Solutions to the questions below will be provided in tomorrow’s blog post, but please post your solutions or partial solutions in the comments below. Basic Definition The first few questions will use the following definition: B is a subset of a set A if every element of B is an element of A. For example, if A = {1,2,3,4,5,6} and B = {4,5,6}, then B is a subset of A. As another example, if C = {a,b}, then all of the subsets of C are { }, {a}, {b}, and {a,b} (here { } is the emptyset, the unique set consisting of no elements). Warm-up Exercises How many subsets does { } have? List them. How many subsets does {1} have? List them. List the subsets of {1,2}, {1,2,3}, and {1,2,3,4}. How many subsets does the set {1,2,3,4,5,6,7,8} have? How many subsets does the set {1,…,n} have where n is a positive integer. Theoretical Exercise Give a proof of the result from problem 5. There are several methods that can be used for number 6. I personally prefer to use Mathematical Induction. Those of you that know this method should give it a try. More information can be found at this Wikipedia page: Mathematical Induction There are several other methods that can be used as well. Please post your attempts at this problem in the comments. Now let’s take a look at some more advanced concepts. Advanced Definitions Define a set to be selfish if the number of elements it has is in the set. For example, X10 = {1,2,3,4,5,6,7,8,9,10} is selfish because it has 10 elements, and 10 is in the set. A selfish set is minimal if none of its proper subsets is also selfish. For example, the set X10 is not a minimal selfish set because {1} is a selfish subset. Let Xn = {1,…,n}. Advanced Exercises How many selfish subsets does Xn have where n is a positive integer? List the minimal selfish subsets of Xn for n = 1, 2, 3, 4, 5, and 6. How many minimal selfish subsets does X10 have? In terms of n, how many minimal selfish subsets does the set Xn have? Give a proof of the result from problem 10. If you liked this article, please share it with your Facebook friends: Solutions will be provided tomorrow. If you would like to begin learning more advanced mathematics, check out my math books Pure Mathematics for Beginners, Set Theory for Beginners, and Topology for Beginners. These books contain no prerequisites and are perfect for anyone just starting out in theoretical math. You can get them all for one low price by clicking on the image below. Comments comments