Overview: Mathematical Language
Conditional statements, also called “if-then” statements are important in logic and mathematics. They state the antecedent as well as the consequent. The conditions are defined precisely in order to determine the parameters of the problem. In order for the entire statement to be correct, both sections must be true.
Conditional Statements
Geometric proofs are usually stated in a shorthand that emphasizes rules and relationships. They use conditional statements, definitions, and postulates in a way that defines the connection between the hypothesis (the “if” portion) and its consequences (the “then” portion). For example, the statement, “If a figure is a square, then it must be a polygon” is only true if the given figure is both a square and that a square is a type of polygon.
Instances and Counterexamples
While one instance, or even thousands of instances, of a statement being true does not necessarily prove it, only one counterexample will disprove it. Suppose the example were turned around such that the conditional statement read “If a figure is a polygon, then it must be a square.” That statement can be disproven with one counterexample. A triangle is a polygon, but it is not a square.
Definitions and Postulates
Before a conditional statement is made, definitions are given. For example, the figure is shown to have four sides of equal length that are perpendicular. In addition, polygons are defined as geometric figures that have three or more sides. If the reader didn’t know that the figure had four sides (the “if” portion), or what the definition of a polygon (the “then” portion, there wouldn’t be a way to connect the parts of the statement. Therefore, there wouldn’t be a way to find instances or counterexamples. Without a definition of the figure, the statement might be true or it might be false. What if the figure really had three sides and angles whose sum equaled 1800, but no one knew it?
Computer Programs
Conditional statements are essential in computer programs. They state the parameters of the problem in such a way that the computer can reach a conclusion. Suppose a computer program is set up to calculate the number of diagonals in any polygon that has more than three sides. To define what the antecedent is, programmers will set up both parts of the conditional statement. If N ≥ 3, then solve N∙(N-3)/2.
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