trying to be a mile wide AND a mile deep
The three properties of relations learned first are reflexivity, symmetry, and transitivity. Reflexivity is an existence property; a possession property. If you contain this entire particular set of pairs, you’re reflexive. If not, then not. Symmetry and transitivity are implications; closure properties. Now some pairs don’t come for free – they require other pairs be […]
There are basically 4 techniques for solving an indefinite integral. Directly/by rules. After algebraic manipulation. Using substitution. By parts. Using limits for improper bounds or internal discontinuities should be mentioned here, but it’s not really in the same category since you don’t use the limit to find an antiderivative. is the method you learn first. […]
Theorem: Suppose a, b, c are natural numbers. If a divides b and b divides c, then a divides c. Bad Proof: Suppose the hypothesis. We can substitute a times something for b in the equation b times something equals c. The two somethings together multiply a to c so we’re done.[way too vague; might […]
Let A = {1,2,3,4,5}. Define a function f from A to A that is also a transitive relation, but is not the identity function. Answer after the jump.
Proof traits explicit/specific (non-vague) logically sound, including complete lacking irrelevant statements understandable to the reader self-contained (may assume basic things; anything else needs explicit reference to previous work or must be written out in the proof) Proof kinds Direct proof: Assume hypothesis and march to conclusion. Contradiction: If proving a single clause, assume its negation […]
I learned this visualization of the chain rule from one of my grad school officemates. It draws on the fact that “and” usually goes with multiplication (more on that in its own post eventually). Imagine when you have f(g(x)) that x is a little sealed box. It is inside a second sealed box marked g, […]