Converse, Inverse, and Contrapositive

By Daniel Machado Updated: 10/11/2026

A conditional statement has the formal structure p → q and is read "if p, then q", where proposition p is called the hypothesis (or antecedent) and proposition q is called the conclusion (or consequent). From this initial relation, three related conditional statements can be formed by altering the order and the truth values of its components: the converse, the inverse, and the contrapositive.

Table of Contents

Comparison Table

To identify how the hypothesis and conclusion change relative to the original conditional statement, their formation rules, symbolic notation, and logical equivalence are summarized in the following table:

Statement	Symbolic Form	Verbal Structure	Formation Rule	Equivalent to p → q
Conditional statement	p → q	If p, then q	Original statement	Yes (by identity)
Converse	q → p	If q, then p	Switch the hypothesis and the conclusion	No
Inverse	¬p → ¬q	If not p, then not q	Negate both components while keeping the original order	No
Contrapositive	¬q → ¬p	If not q, then not p	Switch and negate both components simultaneously	Yes (p → q ≡ ¬q → ¬p)

Although the converse and inverse are not logically equivalent to the original conditional statement, they are logically equivalent to each other (q → p ≡ ¬p → ¬q). When a conditional statement p → q and its converse q → p are both true at the same time, they combine into a biconditional statement, denoted as p ↔ q, which is read "p if and only if q".

Converse Statement

Given an initial conditional statement p → q, its converse is defined as q → p. To form it, switch the order of the components: the conclusion becomes the hypothesis, and the hypothesis becomes the conclusion.

The converse is not equivalent to the original conditional statement. The fact that the conclusion q occurs does not guarantee that the initial condition p caused it, as other circumstances could lead to the same outcome.

To examine how the logical meaning changes relative to the original implication, consider the following examples:

Example 1

Given the statement "if it is raining, then the ground is wet" (p → q), its converse is "if the ground is wet, then it is raining" (q → p).

These statements are not equivalent because the ground could be wet for other reasons, such as a garden hose or a sprinkler, without any rain falling.

Example 2

Given the statement "if a geometric figure is a square, then it is a rectangle" (p → q), its converse is "if a geometric figure is a rectangle, then it is a square" (q → p).

This converse is not equivalent to the original statement. A rectangle has four right angles but may have adjacent sides of different lengths, which prevents it from being a square.

Example 3

Given the statement "if an integer ends in 0, then it is divisible by 5", its converse is "if an integer is divisible by 5, then it ends in 0".

There is no logical equivalence between the two statements. For instance, the number 15 is divisible by 5 but does not end in 0, making the converse false even though the original conditional statement is true.

Example 4

Given the statement "if a triangle is equilateral, then all three of its sides have equal length", its converse is "if all three sides of a triangle have equal length, then it is equilateral".

In this case, both the original conditional statement and its converse are true at the same time. When this happens, both statements are equivalent and can be joined into a biconditional statement: "a triangle is equilateral if and only if all three of its sides have equal length" (p ↔ q).

Inverse Statement

Given a conditional statement p → q, its inverse is formulated as ¬p → ¬q. To form it, negate both the hypothesis and the conclusion while preserving the original order of the conditional.

The inverse is not equivalent to the original conditional statement. Negating the initial condition p does not prevent the outcome q from occurring through a different cause.

Consider the following cases to observe the difference in their truth values:

Example 1

Given the statement "if it is raining, then the ground is wet" (p → q), its inverse is "if it is not raining, then the ground is not wet" (¬p → ¬q).

They are not equivalent because the absence of rain does not ensure that the ground stays dry; it can still get wet from watering a lawn or a water leak.

Example 2

Given the statement "if a triangle is equilateral, then it is isosceles" (p → q), its inverse is "if a triangle is not equilateral, then it is not isosceles" (¬p → ¬q).

This statement is not equivalent to the original. A triangle with side lengths of 5 cm, 5 cm, and 3 cm is not equilateral, but it is still isosceles because it has two equal sides.

Example 3

Given the statement "if an integer is a multiple of 4, then it is an even number", its inverse is "if an integer is not a multiple of 4, then it is not an even number".

These statements are not equivalent. The number 6 is not a multiple of 4, yet it is an even number, which disproves the inverse statement.

Contrapositive Statement

Given a conditional statement p → q, its contrapositive is formally defined as ¬q → ¬p. To form it, switch the order of the components and negate both of them simultaneously.

The contrapositive is logically equivalent to the original conditional statement (p → q ≡ ¬q → ¬p). This means both statements always share the exact same truth value: if the original conditional statement is true, its contrapositive is also true; if the original is false, the contrapositive is also false.

Verify this relationship across the following practical cases:

Example 1

Given the statement "if it is raining, then the ground is wet" (p → q), its contrapositive is "if the ground is not wet, then it is not raining" (¬q → ¬p).

Both statements are logically equivalent. If we confirm that the ground is completely dry, we can directly conclude that it cannot be raining on it.

Example 2

Given the statement "if two angles are vertical angles, then they have the same measure" (p → q), its contrapositive is "if two angles do not have the same measure, then they are not vertical angles" (¬q → ¬p).

There is complete logical equivalence between both statements. If two angles have different measures, it immediately rules out the possibility that they are vertical angles formed by intersecting lines.

Example 3

Given the statement "if n² is even, then n is even" (where n is an integer), its contrapositive is "if n is not even (is odd), then n² is not even (is odd)".

Both statements are equivalent. If we take an odd integer in the form n = 2k + 1, its square is (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is also odd. Because the contrapositive is true, the original conditional statement is automatically proven true.

Proof by Contrapositive

The equivalence p → q ≡ ¬q → ¬p makes the contrapositive an essential tool for indirect proofs. When proving that hypothesis p directly implies conclusion q is algebraically difficult, we assume the negation ¬q and deduce ¬p. Once the contrapositive is proven true, the original conditional statement is proven as well.

Consider the following theorem about integers: "if the product a · b is even, then a is even or b is even". In this statement, we identify three simple propositions:

p: the product a · b is even.
q: a is even.
r: b is even.

The logical structure corresponds to p → (q ∨ r). To form its contrapositive, switch the terms and negate the disjunction using De Morgan's laws, where ¬(q ∨ r) ≡ ¬q ∧ ¬r. Thus, the contrapositive takes the form (¬q ∧ ¬r) → ¬p, which translates into the verbal statement: "if a is odd and b is odd, then the product a · b is odd".

To complete the proof, follow these steps:

Assume that both a and b are odd integers, meaning there exist integers k and m such that a = 2k + 1 and b = 2m + 1.
Multiply the two expressions: a · b = (2k + 1)(2m + 1) = 4km + 2k + 2m + 1.
Factor out a 2 from the first three terms of the expression: a · b = 2(2km + k + m) + 1.
Define the integer t = 2km + k + m, yielding a · b = 2t + 1, which proves that the product is an odd integer (¬p).

Since the contrapositive is proven true, the original conditional theorem is established.

To observe how all three derived statements interact and compare their truth values with the original conditional statement, consider the following four cases.

Example 1

Consider the true conditional statement: "if a person is in Paris, then they are in France" (p → q). From this, we form its three variations:

Converse (q → p): "if a person is in France, then they are in Paris". This statement is false because the person could be in other French cities such as Lyon or Marseille.
Inverse (¬p → ¬q): "if a person is not in Paris, then they are not in France". This statement is false because not being in Paris does not prevent someone from residing in or traveling through another region of France.
Contrapositive (¬q → ¬p): "if a person is not in France, then they are not in Paris". This statement is true because it is impossible to be inside Paris if one is outside French territory.

Example 2

Consider the true geometric conditional statement: "if a geometric figure is a square, then it is a parallelogram" (p → q). We evaluate its logical variations:

Converse (q → p): "if a geometric figure is a parallelogram, then it is a square". This is false because parallelograms can have unequal adjacent sides or non-right angles, such as rhombuses or rhomboids.
Inverse (¬p → ¬q): "if a geometric figure is not a square, then it is not a parallelogram". This is false because figures like rectangles are not squares, yet they remain parallelograms.
Contrapositive (¬q → ¬p): "if a geometric figure is not a parallelogram, then it is not a square". This is true because every square must satisfy the definition of a parallelogram.

Example 3

Consider the true numerical conditional statement: "if an integer is a multiple of 6, then it is a multiple of 3" (p → q). We formulate the related statements:

Converse (q → p): "if an integer is a multiple of 3, then it is a multiple of 6". This is false; for example, the number 9 is divisible by 3 but is not a multiple of 6.
Inverse (¬p → ¬q): "if an integer is not a multiple of 6, then it is not a multiple of 3". This is false; using the number 9 again, we see that it is not a multiple of 6, yet it is a multiple of 3.
Contrapositive (¬q → ¬p): "if an integer is not a multiple of 3, then it is not a multiple of 6". This is true because an integer lacking the prime factor 3 can never contain the factor 6.

Example 4

Consider the true geometric relationship: "if two angles are right angles, then they have the same measure". We examine its three variations:

Converse: "if two angles have the same measure, then they are right angles". This is false because two angles can each measure 45° and have equal measures without being right angles.
Inverse: "if two angles are not right angles, then they do not have the same measure". This is false because two acute angles measuring 30° are not right angles, yet they share the exact same measure.
Contrapositive: "if two angles do not have the same measure, then they are not right angles". This is true because two angles with different measures cannot both measure 90° simultaneously.

Truth Table

To verify the logical equivalence relationships, we evaluate the truth values of all four statements simultaneously across all possible combinations of simple propositions p and q using a truth table:

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

Examining the resulting columns yields the following conclusions regarding validity and logical equivalence:

Equivalence between conditional and contrapositive: the columns for p → q and ¬q → ¬p show identical truth values in every row (T, F, T, T). This formally proves that both statements are logically equivalent (p → q ≡ ¬q → ¬p) and always share the same truth value.
Equivalence between converse and inverse: the columns for q → p and ¬p → ¬q match across all combinations (T, T, F, T). Therefore, the converse and the inverse are logically equivalent to each other (q → p ≡ ¬p → ¬q), making the inverse the contrapositive of the converse.
Lack of equivalence with the original conditional: comparing the column of p → q with those of q → p and ¬p → ¬q in the second and third rows reveals differing truth values. When one is true, the other is false, confirming that the truth of the converse or inverse cannot be deduced from the truth of the original conditional statement.

Practice Problems with Solutions

For each of the following conditional statements, write the converse, inverse, and contrapositive in both symbolic and verbal forms, and determine the truth value of each statement:

If a person has a fever, then their body temperature is above 98.6 °F (37 °C).
If an integer is even, then the remainder when divided by 2 is equal to 0.
If a triangle is a right triangle, then it has two acute angles.
If a polygon is a triangle, then the sum of its interior angles is equal to 180°.
If x = 3, then x² = 9.

Solution 1

Identify the simple propositions that make up the statement: p: "a person has a fever" and q: "their body temperature is above 98.6 °F (37 °C)".

Original (p → q): "if a person has a fever, then their body temperature is above 98.6 °F (37 °C)". This is true, as an elevated body temperature above the standard baseline is the primary clinical indicator of a fever.
Converse (q → p): "if a person's body temperature is above 98.6 °F (37 °C), then they have a fever". This is false; body temperature can rise due to vigorous exercise or heat exhaustion without a clinical fever being present.
Inverse (¬p → ¬q): "if a person does not have a fever, then their body temperature is not above 98.6 °F (37 °C)". This is false; the absence of a fever does not prevent body temperature from rising due to external environmental or metabolic factors.
Contrapositive (¬q → ¬p): "if a person's body temperature is not above 98.6 °F (37 °C), then they do not have a fever". This is true; if the temperature does not exceed the baseline threshold, a fever is ruled out.

Solution 2

Define the simple propositions: p: "an integer is even" and q: "the remainder when dividing the integer by 2 is equal to 0".

Original (p → q): "if an integer is even, then the remainder when divided by 2 is equal to 0". This is true by the definition of integer divisibility.
Converse (q → p): "if the remainder when dividing an integer by 2 is equal to 0, then it is even". This is true, because any integer evenly divisible by 2 belongs to the set of even numbers.
Inverse (¬p → ¬q): "if an integer is not even, then the remainder when divided by 2 is not equal to 0". This is true; an odd integer always leaves a remainder of 1 when divided by 2.
Contrapositive (¬q → ¬p): "if the remainder when dividing an integer by 2 is not equal to 0, then it is not even". This is true; obtaining a non-zero remainder confirms that the integer is odd.

Because the original statement p → q and its converse q → p are both true simultaneously, they form a biconditional statement: "an integer is even if and only if the remainder when divided by 2 is equal to 0" (p ↔ q).

Solution 3

Establish the simple propositions: p: "a triangle is a right triangle" and q: "the triangle has two acute angles".

Original (p → q): "if a triangle is a right triangle, then it has two acute angles". This is true; because it contains a 90° right angle, the remaining two angles must sum to 90°, making each strictly less than 90°.
Converse (q → p): "if a triangle has two acute angles, then it is a right triangle". This is false; an acute triangle has three acute angles (satisfying the condition of having at least two), and an obtuse triangle has two acute angles and one obtuse angle, neither of which are right triangles.
Inverse (¬p → ¬q): "if a triangle is not a right triangle, then it does not have two acute angles". This is false; an oblique triangle still contains two acute angles within its interior structure.
Contrapositive (¬q → ¬p): "if a triangle does not have two acute angles, then it is not a right triangle". This is true; if a triangular figure does not possess at least two acute angles, it cannot satisfy the angle requirements of a right triangle.

Solution 4

Consider the propositional components: p: "a polygon is a triangle" and q: "the sum of the polygon's interior angles is equal to 180°".

Original (p → q): "if a polygon is a triangle, then the sum of its interior angles is equal to 180°". This is true by the triangle angle sum theorem in Euclidean geometry.
Converse (q → p): "if the sum of the interior angles of a polygon is equal to 180°, then it is a triangle". This is true; the interior angle sum formula 180° · (n − 2) equals 180° only when the number of sides is n = 3.
Inverse (¬p → ¬q): "if a polygon is not a triangle, then the sum of its interior angles is not equal to 180°". This is true; any polygon with 4 or more sides has an interior angle sum of 360° or greater.
Contrapositive (¬q → ¬p): "if the sum of the interior angles of a polygon is not equal to 180°, then it is not a triangle". This is true; a polygon whose angle sum differs from 180° cannot be a triangle.

Because the conditional statement and its converse are true simultaneously, we can state the biconditional statement: "a polygon is a triangle if and only if the sum of its interior angles is equal to 180°" (p ↔ q).

Solution 5

Define the algebraic propositions: p: "x = 3" and q: "x² = 9".

Original (p → q): "if x = 3, then x² = 9". This is true, since squaring 3 yields 9.
Converse (q → p): "if x² = 9, then x = 3". This is false; the quadratic equation also allows the solution x = −3, since (−3)² = 9.
Inverse (¬p → ¬q): "if x ≠ 3, then x² ≠ 9". This is false; if we let x = −3, it is not equal to 3, yet its square is still 9.
Contrapositive (¬q → ¬p): "if x² ≠ 9, then x ≠ 3". This is true; if the square of a number is not 9, that number cannot be 3.