Understanding Truth Tables and Boolean Logic Fundamentals
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Truth tables are useful logical tools found in mathematics, computer science, and philosophy, making them valuable across various disciplines. While the notation may differ depending on the field, the underlying principles remain consistent.
This introductory guide will provide you with the foundational knowledge necessary to grasp symbolic logic. We will begin by defining key operators, and in the following post, I will demonstrate how to analyze a more complex logical statement.
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What is Boolean Algebra?
Boolean Algebra is a segment of algebra that deals with Boolean values, specifically true and false. These values are typically represented as T (or 1) for true and F (or 0) for false. This simple system allows us to simplify complex statements into manageable logical expressions.
Unary Operators
Unary operators are the most straightforward operations as they operate on a single True or False value.
Identity
The identity operator is the simplest case, affirming that True is True and False is False.
Negation
The negation operator, often denoted by the tilde (~) or ¬ symbol, inverts the truth value of a statement.
This relationship can be illustrated using a truth table, which organizes information to display all potential scenarios. We label the first column as "p" for proposition, and in the second column, we apply the operator to p (in this case, ~p). Thus, if our initial premise is True and we negate it, we obtain False, and vice versa.
Logical True and Logical False
These operations behave quite uniquely. Logical true consistently results in True, while logical false consistently yields False, regardless of the premise. They are often labeled as “always true” and “always false.”
Binary Operators
Binary operators work with two propositions, which we will denote as p and q.
AND
The AND operator (symbol: ?), also known as logical conjunction, requires both p and q to be True for the outcome to be True; all other combinations yield False. This mirrors the intersection of two sets in a Venn Diagram.
OR
The OR operator (symbol: ?) only requires one of the premises to be True for the result to also be True, akin to the union of two sets in a Venn Diagram.
NOR
The NOR operator (symbol: ?) is the negation of OR. It requires both p and q to be False for the result to be True.
XOR
Exclusive OR, or XOR (symbol: ?), requires exactly one True and one False value for the outcome to be True.
Conditional Operators
Implication
Logical implication (symbol: p ? q), also known as "if-then", results in True in all scenarios except when T ? F. To aid memory, this can be viewed as logically equivalent to ¬p ? q (read: not p or q).
To illustrate their equivalence, we can create a truth table. Begin with the usual p and q columns, then add a "¬p" column reflecting the opposite truth values of p. Finally, compute ¬p ? q by applying the OR operation to the second and third columns.
It’s crucial to recognize that ¬p ? q ? ¬(p ? q). In the former, p is negated, while in the latter, the resultant truth value of (p ? q) is negated.
Logical Equality
Also referred to as the biconditional or if and only if (symbol: ?), logical equality is expressed as (p ? q) ? (q ? p). In simpler terms, it’s an if-then statement where the converse is also valid.
The only way to affirm that a conditional is valid in both directions is if both p and q share the same truth value, meaning they are either both True or both False. This is why the biconditional is often termed logical equality.
Surprisingly, these definitions cover most of the logic problems you will encounter.
In the next article, I will demonstrate how to apply these definitions to create a truth table for a logical expression like (A ? ~B) ? (C ? D).
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