Definition of a Mathematical Statement
A mathematical statement is a declarative sentence that is either definitively true or definitively false, but not both. It must have a truth value that can be objectively determined. Unlike expressions, which represent a value, statements express a complete thought that can be judged for its veracity. For instance, '2 + 2 = 4' is a true statement, while '2 + 2 = 5' is a false statement.
Key Characteristics and Types
Mathematical statements can be simple or compound. Simple statements are basic assertions without logical connectives. Compound statements combine two or more simple statements using logical operators like 'and' (conjunction), 'or' (disjunction), 'not' (negation), 'if...then' (implication), and 'if and only if' (biconditional). For example, 'All squares are rectangles' is a simple statement, and 'If a number is even, then it is divisible by 2' is a compound (conditional) statement.
Statements vs. Expressions
It's crucial to distinguish statements from expressions. An expression is a phrase that represents a value, such as '2x + 3' or 'y^2'. It doesn't have a truth value itself because its value depends on the variables. Only when an expression is equated to something or asserted (e.g., '2x + 3 = 7' or '2x + 3 > 5') does it become part of a mathematical statement that can be true or false.
Importance in Mathematical Proofs
Mathematical statements are the building blocks of logical arguments and proofs. Every step in a proof relies on establishing the truth of a sequence of statements, ultimately leading to the conclusion. Understanding how to construct and evaluate mathematical statements is fundamental to reasoning rigorously, formulating conjectures, and verifying mathematical theorems across all branches of mathematics.