Contrapositive

Contrapositive Statements in Geometry Explained with Examples

contrapositiveWhat is Contrapositive?

Contents

Definition: Contrapositive is exchanging the hypothesis and conclusion of a conditional statement and negating both hypothesis and conclusion.

For example the contrapositive of “if A then B” is “if not-B then not-A”. The contrapositive of a conditional statement is a combination of the converse and inverse.

Conditional statement: A conditional statement also known as an implication. A conditional statement is in the form “If p, then q” where p is the hypothesis while q is the conclusion.


Contrapositive Statement Characteristics

  • The contrapositive of any true proposition is also true.
  • Contrapositive of a true statement is also true.
  • Contrapositive of a false statement is also false.

The Law of Contrapositive in Geometry

The conditional statement and its contrapositive are logically equivalent.

Uses

  • Contrapositive is used when an implication has many hypotheses or when the hypothesis specifies infinitely many objects.
  • Contrapositive can be used as a strong tool for proving mathematical theorems because contrapositive of a statement always has the same truth table.

Contrapositive Formula

If the conditional of a statement is p q then, we can compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement. The contrapositive of p q is q p.

The contrapositive of a conditional statement is a combination of the converse and inverse.

If the conditional statement is p q

Converse statement of p q is q p.

The hypothesis p of the conditional statement becomes the conclusion of the converse.

The conclusion q of the conditional statement becomes the hypothesis of the converse.

Inverse statement of q p is q p.

The inverse statement is obtained by negating both hypothesis and conclusion.

‘If q then p’ is a contrapositive of the conditional statement ‘if p then q’.

Contrapositive of a conditional statement is logically equivalent to its conditional statement.


Conditional Statement Examples

Conditional statement:

If it is raining, then the grass is wet.
The contrapositive of this statement is:
If the grass is not wet then it is not raining.

Conditional statement:

If a figure is a square then all the four sides are equal.
The contrapositive of this statement is:
If all the four sides are not equal then it is not a square.

Conditional statement:

If x is equal to zero, then sin(x) is equal to zero.
The contrapositive of this statement is:
If sin(x) is not zero, then x is not zero.

Conditional statement:

If am standing in Manchester, then I am standing in United Kingdom.
The contrapositive of this statement is:
If I am not standing in United Kingdom, then I am not standing in Manchester.

Conditional statement:

If a polygon is a triangle, then it has 3 sides.
The contrapositive of this statement is:
If the polygon does not have three sides, then it is not a triangle.


Contrapositive Truth Table example

Conditional v/s Contrapositive

p: If it rains

q: they cancel school

        Conditional Converse Inverse Contrapositive
p q p q p q q p p q q p
T T F F T T T T
T F F T F T T F
F T T F T F F T
F F T T T T T T

 

If the conditional statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.

Conditional Statement If p, then q p q
Converse If q, then p q p
Inverse If not p, then not q p q
Contrapositive If not q, then not p q p

Summary:

The contrapositive of a conditional statement is the mixing of the converse and inverse.

“If the sun sets down” is the hypothesis

“It’s in the west” is the conclusion.

To form the converse of the conditional statement, interchange the hypothesis and the conclusion.

The converse of “The sun sets down then it’s in the west” is “The sun is in the west, if it sets down”.

To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.

The inverse of “The sun sets down then it’s in the west” is “The sun is not in the west, if it not sets down”.

To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.

The contrapositive of “The sun sets down then it’s in the west” is “The sun not sets down, if it is not in the west”.

Example:

Statement If two angles are congruent, then they have the same measure. (True)
Converse If two angles have the same measure, then they are congruent. (True)
Inverse If two angles are not congruent, then they have not the same measure. (True)
Contrapositive If two angles do not have the same measure, then they are not congruent. (True)

 

Statement If a figure is a rhombus, then its diagonals are perpendicular. (True)
Converse If diagonals are perpendicular, then it is rhombus. (False)
Inverse If a figure is not a rhombus, then its diagonals are not perpendicular.(False)
Contrapositive If diagonals are not perpendicular, then it is not rhombus. (True)