Propositional Logic gives us tools for precisely describing and analyzing logical statements. Real-world problems are usually given to us in English. In order to apply our new tool, we must first “formalize” the problem by translating the statement into Propositional Logic. Then, we can analyze it using tools such as truth tables or SAT solvers.
In this next topic, we will extend our knowledge of logic in a few ways. First, we will show the connection between Propositional Logic and circuits. Second, we will look at how to prove logical expressions (or circuits) equivalent without truth tables. Finally, we will extend our logic with the ability to talk about properties of objects, giving us Predicate Logic.
In this topic, we will learn a new way to reason about facts in logic. Previously, we saw how to show that various facts are equivalent. Here, we will see how to infer new facts from known ones. This includes equivalence as a special case (when we can infer either fact from the other) but is vastly more powerful.
We will begin by writing our proofs formally, which makes them easy to check. Formal proofs are built up using inference rules. Then, we will begin learning how to write English proofs by translating our formal proofs into English. As the course continues, we aim to get more confortable writing proofs directly in English, without working formally.
In the remainder of the course, we will look at different kinds of mathematical objects that show up frequently in computer science. In addition to learning about their properties, this will give us settings in which to practice the proof techniques we learned in Topic 3.
In this topic, we look at important properties of numbers. All data in a computer is stored as numbers and the only operations computers can do on their own are arithmetic calculations. For that reason, numbers arise everywhere in computer science, and we need to understand them well in order to make computers do interesting things for us.
In this next topic, we will extend our knowledge of number theory in a few ways. First, we will use our knowledge of congruences to show that we can solve for the variable that appears in a congruence (here, usually called a “modular equation”). Second, we will add two new inference rules that apply specifically to the domain of non-negative integers. These allow us to prove for-all claims over the non-negative integers in a new way. Finally, we will look at a few more applications of number theory that arise widely in computer science.
More topics coming as the course continues...