Counting Kings

(A new problem of the week) Having discussed counting earlier this week, let’s take a look at a different kind of counting. The subject of combinatorics (the study of counting) arises in many guises: probability, sets, geometry. Here, we look at a relatively basic type of problem that involves the same sort of organized thinking …

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Venn Diagrams: Language Issues

(A new question of the week) I mentioned that we have had a number of questions related to Venn diagrams recently. Here I would like to show a couple of these, from a Philippine student. Even fluent English speakers can get confused in these problems; observing how a student new to the language misinterprets details …

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Venn Diagrams: Over the Top

(An archive question of the week) Last time we looked at various 2- and 3-set Venn diagram problems (and alternative methods). One discussion I found while collecting them deserved to be set aside for special examination, if only because it would scare the beginner. A mixture of tools will make the work easier. Here it …

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Properties as Axioms or Theorems

To close out this series that started with postulates and theorems in geometry, let’s look at different kinds of facts elsewhere in math. What is commonly called a postulate in geometry is typically an axiom in other fields (or in more modern geometry); but what about those things we call properties (in, say, algebra)?

Subtleties in a Logic Puzzle

(Archive Question of the Week) Logic puzzles can exercise our ability to reason carefully. Interestingly, the use of formal logic in doing so can actually get in our way, because such puzzles often have subtleties in their wording that are hard to capture in formal logic. Examining our thinking carefully can help us see wrong …

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Empty Sets and Vacuous Truth

I’m going to start this post with a simple question about the empty set, and gradually dive deeper. There will be connections here to previous discussions of conditional statements in logic.