Fundamentals of Math

Fundamentals of Mathematics

Chapter 8: Logic Our final topic in this book will be an introduction to the field of logic as it relates to mathematics. Logic is a vital aspect of math, as it provides a structure and foundation to our number systems. This chapter will introduce you to the basic concepts and symbols of Boolean Algebra - a branch of algebra that involves bools, or true and false values. Mathematicians usually subscribe to the Law of the Excluded Middle , which states that for any proposition, either that proposition is true or false. In the study of logic, these propositions are referred to as sentences, but be careful, a sentence in mathematics is not the same definition as a sentence in an English course. In mathematics, sentences are a mathematical or logical statement (such as an equation or a proposition) in words or symbols. Sentences can be negated or combined to create even more complex sentences, but at its core, every sentence is categorized into one of two values - true (T) or false (F). Note: Sometimes, you may see the binary system used in place of T and F. In this instance, 1 is equivalent to true, and 0 is equivalent to false. To determine if a statement is a sentence, you must be able to assess if the statement is true or false. For example, "Look out!" is a sentence in the English language, but we cannot evaluate it (true or false) and therefore is not a sentence in the mathematical sense. Whereas, "A daily surplus of calories will cause a person to gain weight" can be evaluated as a true or false claim. The same idea can be applied to mathematical equations. Such as, " + 3 = 4 " is not a sentence because we cannot evaluate if that statement is true or false, whereas " + 3 = 4 when = 1 " is a sentence because we can substitute the value of 1 into in the equation and evaluate the equation (true or false). For each sentence, we can apply unary or binary operators. Unary operators are what negate a sentence (e.g., I will not go to the store or ≠ 2 ). Some of the most common binary operators used are the conjunction, disjunction, conditional, or bi-conditional. Conjunctions essentially combine sentences using the "and" operator, whereas disjunction combine sentences using the "or" operator. The conditional operator combines sentences using the "if this then that" format, which is similar to the "if and only if" format used by the biconditional operator.

and are typically used to denote two independent sentences.

Operation Conjunction Disjunction Conditional

Meaning

Symbol Example

And

∧ ∨

∧ ∨

Or

If … then …

Biconditional … if and only if* …

Negation

Not

~

~

* … if and only if is sometimes denoted iff

Now, although and could have been true or false on their applying a unary or binary operator to one or both of them would cause us to have to re-assess their validity.

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