High School Statutory Authority: Algebra I, Adopted One Credit.
Examples[ edit ] Take the statement "All red objects have color. The inverse is "If an object is not red, then it does not have color. Therefore, in this case the inverse is false.
The converse is "If an object has color, then it is red. The negation is "There exists a red object that does not have color.
In other words, the contrapositive is logically equivalent to a given conditional statement, though not sufficient for a biconditional. Similarly, take the statement "All quadrilaterals have four sides," or equivalently expressed "If a polygon is a quadrilateral, then it has four sides.
The inverse is "If a polygon is not a quadrilateral, then it does not have four sides. 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. This contrapositive, like the original statement, is also true. The latter can be proved by contradiction. The previous example employed the contrapositive of a definition to prove a theorem.
One can also prove a theorem by proving the contrapositive of the theorem's statement. To prove that if a positive integer N is a non-square numberits square root is irrational, we can equivalently prove its contrapositive, that if a positive integer N has a square root that is rational, then N is a square number.
Correspondence to other mathematical frameworks[ edit ] Contraposition represents an instance of Bayes' theorem which in a specific form can be expressed as:a) Describe at least five different ways to write the conditional statement p → q in English.
(Solved) August 02, a) Describe at least five different ways to write the conditional statement p → q in English. b) Define the converse and contrapositive of a conditional statement. Jul 02, · To write these statements in if -then form, identify the hypothesis and conclusion. The word if is not part of the hypothesis.
The word then is not part of the conclusion. If the angle is acute, then its measure is between 0 and Write the converse, inverse, and contrapositive of each true conditional statement. 5 CONTRAPOSITIVE: switch and negate the hypothesis and conclusion of a conditional statement.
Conditional: If I don't do my homework, then I will get.
State the converse, contrapositive, and inverse of each of these conditional statements. a) If it snows tonight, then I will stay at home.
b) I come to class whenever there is going to be a quiz. Resource Locker GB Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement and recognize the connection between a biconditional statement and a true conditional statement with a true converse.
5. For each conditional statement given, write the converse, inverse, and contrapositive.
Then determine the truth value of each statement. If a statement is false, give a counterexample. a.
If a point is in the first quadrant, then both its coordinates are positive. converse: inverse: contrapostive: b. If you are in Chicago, then you are in Illinois.