Logical Operations and Truth Tables

At first glance, it may not seem that the study of logic should be part of mathematics. For most of us, the word logic is associated with reasoning in a very nebulous way:

"If my car is out of gas, then I cannot drive it to work." seems logical enough, while
"If I am curious, then I am yellow." is clearly illogical. Yet our conclusions about what is or is not logical are most often unstructured and subjective. The purpose of logic is to enable the logician to construct valid arguments which satisfy the basic principle
"If all of the premises are true, then the conclusion must be true."
It turns out that in order to reliably and objectively construct valid arguments, the logical operations which one uses must be clearly defined and must obey a set of consistent properties. Thus logic is quite rightly treated as a mathematical subject.
Up until now, you've probably considered mathematics as a set of rules for using numbers. The study of logic as a branch of mathematics will require you to think more abstractly then you are perhaps used to doing. For instance, in logic we use variables to represent propositions (or premises), in the same fashion that we use variables to represent numbers in algebra. But while an algebraic variable can have any number as its value, a logical variable can only have the value True or False. That is, True and False are the "numerical constants" of logic. And instead of the usual arithmetic operators (addition, subtraction, etc.), the logical operators are "AND", "OR", "NOT", "XOR" ("eXclusive OR"), "IMPLIES" and "EQUIVALENCE". Finally, rather than constructing a series of algebraic steps in order to solve a problem, we will learn how to determine whether a statement is always true (a tautology) or is a contradiction (never true), and whether an argument is valid.

Truth Tables

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