If -Then Statements and Postulates

If–then statements can be used to clarify statements that seem confusing. Consider this example: The following statement is part of a message frequently played by radio stations across the nation. "If this had been an actual emergency, then the attention signal you just heard would have been followed by official information, news, or instruction." This statement is an "if–then" statement because there is a hypothesis and there is a conclusion. The hypothesis of the statement is the section following directly after the "if" (If this had been an actual emergency,). The conclusion of the statement is the section following directly after the "then" (then the attention signal you just heard would have been followed by official information, news, or instruction.) Put all together, it is an if–then statement because it has a hypothesis and a conclusion.

If–then statements are also called conditional statements or conditionals. Conditional statements include the words "if" and "then" in the statements, representing the hypothesis and the conclusion.

It is possible to form another "if–then" statement by exchanging the hypothesis and conclusion of a conditional. This new statement is known as a converse. The converse of P–Q is Q–P. The converse of If two lines are perpendicular, then they intersect is If two lines intersect, then they are perpendicular. It may be simpler to write a conditional in "if–then" form first before writing the converse. The converse of a true statement is not necessarily true and thus can be proven false.

The denial of a statement is called a negation.  An example of a negation of If you are a pilot, then you fly airplanes is If you aren't a pilot, then you don't fly airplanes. If a statement is true, then its negation is false. If a statement is false, then its negation is true.

A conditional statement has an inverse, which can be formed by negating both the hypothesis and the conclusion. The inverse of a true statement is not necessarily true and can be proven false.

The inverse of this statement is false because a square, which is not a triangle, is a polygon.

Postulates are principles that are accepted to be true without proof. The following postulates describe ways that points, lines, and planes are related. 

Postulate 2-1: Through any two points, there is exactly one line.

Postulate 2-2: Through any three points not on the same line, there is exactly one plane.

Postulate 2-3: A line contains at least two points.

Postulate 2-4: A plane contains at least three points not on the same line.

Postulate 2-5: If two points lie in a plane, then the entire line containing those two points lies in that plane.

Postulate 2-6: If two planes intersect, then their intersection is a line.

Resources:

Geometry Inductive and Deductive Reasoning. 14  October 2011. <http://en.wikibooks.org/wiki/Geometry/Inductive_and_Deductive_Reasoning>.

Geometry. 14 October 2011. <http://library.thinkquest.org/28586/640x480x8/glossary/>.

Glossary. 14 October 2011. <http://www.learner.org/courses/learningmath/geometry/keyterms.html>.

Boyd, Cindy J., Burril, Gail F., Cummins, Jerry J., Kanold, Timothy D., Malloy, Carol. Yunker, Lee E. Gencoe Geometry. Westerville, Ohio: McGraw-Hill Companies Inc., 1998.

Aaron Nolan