Mathematical Logic Basics - Visual Guide

Welcome to Mathematical Logic! Today we'll explore statements and truth values.A statement is a declarative sentence that must be either true or false, but it cannot be both.Let's look at some mathematical examples. Two plus two equals four is true, while five is greater than seven is false.Statements can also come from everyday life. For example, 'The Earth is round' is true, while 'Dogs can fly' is false.In mathematical logic, we represent truth values using symbols. T or 1 represents true, while F or 0 represents false.However, not all sentences are statements. Questions and commands cannot be statements because they don't have truth values.Let's practice identifying statements. Remember, a statement must be a declarative sentence that can be either true or false.The AND operator, symbolized by ∧, requires both statements to be true for the result to be true.Looking at the truth table, we can see that AND only gives true when both inputs are true.The OR operator, shown as ∨, is true when at least one statement is true.The OR truth table shows that only when both statements are false do we get a false result.The NOT operator, written as ¬, simply reverses the truth value of a statement.As we can see in this simple truth table, NOT turns true to false, and false to true.In mathematical logic, conditional statements express relationships between two statements using if-then structure.The first part is called the hypothesis, and the second part is the conclusion. Together they form an implication.To understand when an implication is true, we use a truth table. Let's examine all possible combinations.Let's look at some real-world examples of conditional statements.Every conditional statement has an equivalent form called its contrapositive.The contrapositive is formed by negating both parts and switching their order. It always has the same truth value as the original statement.This concept is particularly useful in mathematical proofs. Let's look at a mathematical example.Sometimes it's easier to prove a statement using its contrapositive form.