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@jemoka / Jemoka Knowledge Base / raw/concept/kbhproof_design_patterns.md
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--- title: "Proof Design Patterns" source: https://www.jemoka.com/posts/kbhproof_design_patterns/ --- Based on the wise words of a crab, I will start writing down some Proof Design Patterns I saw over Axler everything. inheriting properties (splitting, doing, merging) “complex numbers inherit commutativity via real numbers” construct then generalize for uniqueness and existence try to remember to go backwards to prove IFF zero is cool, and here too!, also \(1-1=0\) \(0v = 0\) \(1-1 = 0\) \(v-v=0\) a.k.a. \(v+(-v)=0\) \(v+0 = v\) distributivity is epic: it is essentially the only tool to connect scalar multiplication and addition in a vector space “smallest” double containement proofs to show set equivalence: prove one way, then prove the converse (\(a \subset b, b\subset a \Rightarrow a=b\)) couple hints step 1: identify hypothesis (assumptions) desired conclusion (results, trying/to/proof) step 2: define write down precise, mathematical notations proving uniqueness: set up two distinct results, show that they are the same proving negation: if the “negative” is distinct, but the direct case is more nebulous, use proves by contradiction proof by induction especially if you are dealing with polynomials, try factoring tools to help includes length of linearly-independent list \(\leq\) length of spanning list Uniqueness by construction: uniqueness part of basis of domain pick one element that does exist pick arbitrary elements and construct a result if we are trying to prove equivalence, double-containment is a good bet see fundamental theorem of linear maps: but basically wehnever you need to construct basis of things start with an arbiturary basis of the subspace and expand into that of the whole space a loop in the statements makes them all equivalent working with the square of the norm is often easier using NFA in place of a DFA can make your life a lot easier; if you need a DFA in the end then just use subset construction remember the NFAs that are useful to make each construction for the properties of regular languages; then, construct any NFA you need to support a particular language out of those properties