Unit 3: Mathematical Logic
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:
- describe declarative statements in math that have 100% "true" or 100% "false" values;
- classify sequences of logical statements as yielding true or false conclusions, given the sequence of statements and the values of individual statement parameters (arguments);
- write a logic equation based on the natural-language statement of a problem; and
- calculate logical expressions of multiple variables assigned values according to a given circumstance.
3.1: Propositions and Logical Operators
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.
3.2: Truth Tables and Propositions Generated by a Set
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.
3.3: Equivalence and Implication
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.
3.4: The Laws of Logic
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.
Unit 3 Assessment
- Receive a grade
Take this assessment to see how well you understood this unit.
- This assessment does not count towards your grade. It is just for practice!
- You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.