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''' Did you know?   A few hack, cracks and obfuscated examples, which may actually lead to less confusion and a greater knowledge of why things work and how. However, these examples have a purpose, and should not be taken, in any way as correct or just another way to achieve a common goal. They are purposely, in some cases, convoluted.   I. String, things, hacking, and unpacking '''   import string s = ‘The quick brown fox jumps over the lazy dog”   The string above (s) is called a pangram. This means it contains every letter of the alphabet at least once. Let’s  Prove that’s true.   Goal: To extract the alphabet in lexicographical form from the pangram below.   alphabet = string.ascii_lowercase # → ‘abcdefghijklmnopqrstuvwxyz’   #Algorithm 1. We begin by calling method lower() on our string to create all lowercase letters (A small amount of normalization).   s... read more

Containment and Equality If A and B are sets, then A is said to be contained in B iff (if and only if) every element of A is contained in B. So A⊆B means that A is a subset of B. Example: All squares ⊆ all rectangles All right triangles ⊆ all triangles Important! This implies the idea of forwards and backwards logic: If Joe has three million dollars, he is a millionaire. If Joe is a millionaire it doesn’t necessarily mean he has three million dollars, he could have one million dollars and still be a millionaire. Likewise, all squares are rectangles but not all rectangles are squares. A=B iff A⊆B and B⊆A Example: {x:x^2=4}={-2,2} {x:x^2<4}={x:-2<x<2}

Sets and Other Elementary Subjects Sets are a collection of things called objects. Objects are all unambiguously defined. In other words, objects have unmistakably clear definitions with one meaning and one interpretation that leads to one conclusion. This may seem convoluted because we are so used to words and phrases having different meanings and whatnot, but not in this case. Look at some examples to get a better idea what it means for objects to be unambiguously defined. Objects                       Not Objects Cars                            Cool cars Children                      Nice children Temperature               Comfortable temperature Baseball players       ... read more

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