In this unit, we will discuss basic concepts that are part of the foundation of mathematical logic. As you seek to fully understand these concepts, you must be able to recognize valid logical arguments. Although these arguments will usually be applied to mathematics, they are the same techniques used by lawyers in the courtroom, physicians examining a patient, or engineers trying to solve a difficult problem. The circuits of computers are designed using the same algebra of propositions that we will discuss in this unit. Often, embedded computers (that is, the data processing units within a larger machine) are programmed by using binary, gate-level logic, machine code, or ladder logic. These rely on the basic concepts we will discuss here.
Completing this unit should take you approximately 5 hours.
Upon successful completion of this unit, you will be able to:
Within this subunit, we encounter basic definitions and operators. Fundamental symbology is also presented and discussed. There are several examples that help you understand the math in terms of human language. You will see several ways to say the same thing, while remaining logically and mathematically correct.
Read these sections to supplement your understanding of propositional logic.
Be sure to review this notation summary since these terms will be used throughout this unit.
What is in a set and what is not in a set leads to some interesting ways of analyzing truth or falsehood. In this section we use 0 for false (no) and 1 for true (yes). One can also speak in terms of "do-not" or "do", "do not perform this action" or "do this action". It is a matter of interpretation, an interpretation that must be established and remain consistent. We can write equations to express these ideas so that many factors can be considered and operated upon in a standard way. This section starts you down that path.
There are many ways to write the same logical equation. Too, various equations are implied by other equations or are contradicted. This section explores that idea. For instance, changing the terminology used to describe an idea, object, or field of study does not change those from what they are. An example are cloud providers who reinvent basic terminology in distributed systems so that their offering appears to be new and unique. An understanding of this area of thought will allow you to see through marketing hype and to simplify logical equations. Thus, you can bring clarity to your understanding and lower costs to systems described by logical expressions.
We will now prepare for the unit on proofs. Essentially, a table of laws is presented and discussed. These are essential to our future study in this topic area. You will find a similarity between laws of logic and laws of algebra. However, just as similarities between the syntax of computer languages can lead you astray, be sure you keep logic and algebra separate. For instance, 1 + 1 does not equal 2 in logic. Rather, 1 + 1 = 1.
Take this assessment to see how well you understood this unit.