Three individuals will be chosen at the beginning of each class to update these three daily blog categories.

Students updating the “Definitions” category, should take the opportunity to engage with important definitions from the weekly readings and lectures. To truly engage with a definition, you are expected to go beyond simply copying down what is written in the textbook. Below are a few suggestions for engaging with definitions that you may apply to this daily blog series:

• “How to Prove ____ Template”: The most important insight to take away from a stated definition is the structure of a proof that something satisfies the given definition. Take relevant definitions from the day’s lecture and come up with a template for what a proof should look like. For example: “Upper Bound: s is an upper bound of the set A if for all .” Proof template: “Let , then …. from which it follows that . Thus, s is an upper bound of A, since was taken arbitrarily.”
• “How to Disprove ____ Template”: To truly test your comfort with a definition, try to enumerate all of the ways one can go about disproving a definition (similar to above, but to prove something does NOT satisfy a given definition). It is more often the case that there are numerous ways to negate a logical statement, so try to be as thorough as possible.
• “Compare and Contrast Definitions”: Many of the definitions we encounter in this class will be slight variations on previous definitions. To fully grasp the distinctions and relations between definitions, it is helpful to explore examples that satisfy one definition and but fail to satisfy another, or try to elaborate (in words) the consequences of slight changes in the logical structure from one definition to the next.
• “Exploring Definition Structure in Other Contexts”: To untangle the often perplexing combination of logical quantities in a definition (implications, quantifiers, negations, etc), it can be helpful to replace the mathematical objects referred to in the definition with everyday objects. For example: To better understand the relation between s and the set A, where s is an upper bound for A, one might consider defining “Ancestor of: Person 1 is an ancestor of People (group of individuals) if for all Individual in People, there exists an unbroken chain of lineage from Person 1 (and his/her offspring) to Individual.”
• That’s all the examples I have at the moment, but feel free to get creative here. The only wrong answers here are ones that either ONLY include definitions directly from the lecture / textbook, or DO NOT contain anything about the definitions relevant to the class whatsoever…