Contraposition - Wikipedia
Conditional statements have special variations that are closely related to the conditional statement in some way. One of those variations is the contrapositive of a. Note: Many students find it helpful to diagram conditional statements, and we . This logically equivalent statement is sometimes called the contrapositive of the These are the two, and only two, definitive relationships that we can be sure of. Conditional statements are so important because they express a relationship between two events. There are several ways to describe the two events linked in a.
Conditional reasoning and logical equivalence (article) | Khan Academy
The converse is "If a polygon has four sides, then it is a quadrilateral. The negation is "There is at least one quadrilateral that does not have four sides.
Since the statement and the converse are both true, it is called a biconditionaland can be expressed as "A polygon is a quadrilateral if, and only if, it has four sides. That is, having four sides is both necessary to be a quadrilateral, and alone sufficient to deem it a quadrilateral. If a statement is true, then its contrapositive is true and vice versa.
If a statement is false, then its contrapositive is false and vice versa. If a statement's inverse is true, then its converse is true and vice versa. If a statement's inverse is false, then its converse is false and vice versa. If a statement's negation is false, then the statement is true and vice versa. If a statement or its contrapositive and the inverse or the converse are both true or both false, it is known as a logical biconditional.
Application[ edit ] Because the contrapositive of a statement always has the same truth value truth or falsity as the statement itself, it can be a powerful tool for proving mathematical theorems. A proof by contraposition contrapositive is a direct proof of the contrapositive of a statement. This statement is true because it is a restatement of a definition.Converse, Inverse, and Contrapositive: Lesson (Geometry Concepts)
All of the points within the inner circle match the premise A. Because the circles are nested, those same points are within circle B they possess the property associated with B.
If-then statement (Geometry, Proof) – Mathplanet
Notice that there are points in circle B that are not inside of circle A. Therefore, the converse statement, if B, then A, does not have to be true. If the converse were true, then circle B would need to be contained within A as well and the two circles would have to be identical. The same points that show that the converse might be false, also show that the inverse is suspect.
There might be examples which do not have property A, but which do have property B so if not A, then not B is not dependable.
- If-then statement
The Euler Diagram for "if A, then B" The Euler Diagram for "if a figure is a triangle, then it is a polygon" A third transformation of a conditional statement is the contrapositive, if not B, then not A. The contrapositive does have the same truth value as its source statement.
Conditional reasoning and logical equivalence
The Euler diagram illustrates why the contrapositive is equivalent to the original statement. Because circle A is wholly within circle B, points outside of circle B not B must be outside of circle A not A as well. So not B implies not A. The equivalence of a statement and its contrapositive is at the heart of the method of proof by contradictionwhich proves that the contrapositive of a conjecture is true and, therefore, that the original conjecture is true.
Freeman has a thorough discussion of these ideas with many good exercises.