Chapter 2 · Reasoning and Proof

Converse, Inverse, Contrapositive, and Biconditionals

Learn to rearrange a conditional into its converse, inverse, and contrapositive, decide which are true, and join two directions into a biconditional.

20 min read

You already know how to write a conditional in if-then form and how to find its truth value, which tells you whether it is true or false. Now you will rearrange its parts. When you swap the hypothesis and the conclusion, or say the opposite of each, you get a new statement that makes its own claim. One of these new statements always matches the original in truth value. The other two may not. In this lesson, you will learn which is which and why.

Call the hypothesis $p$ and the conclusion $q$ , so the conditional is $p \to q$ , read "if $p$ , then $q$ ." The negation of a statement says the opposite of it. You write it as ${\sim}p$ and read it as "not $p$ ." When $p$ is true, ${\sim}p$ is false, and when $p$ is false, ${\sim}p$ is true.

Here is a geometry example. An acute angle is an angle whose measure is greater than $0^\circ$ and less than $90^\circ$ . The negation of "the angle is acute" is "the angle is not acute." Angle measures are always positive, so this negation means the measure is $90^\circ$ or more.

Three new statements built from a conditional

Statement	How to build it	Symbols	How to read it
Conditional	Start with the original statement.	$p \to q$	If $p$ , then $q$ .
Converse	Swap the hypothesis and the conclusion.	$q \to p$	If $q$ , then $p$ .
Inverse	Negate the hypothesis and the conclusion, and keep their order.	${\sim}p \to {\sim}q$	If not $p$ , then not $q$ .
Contrapositive	Swap the hypothesis and the conclusion, then negate both.	${\sim}q \to {\sim}p$	If not $q$ , then not $p$ .

Worked example

Writing the three related statements

Write the converse, inverse, and contrapositive of this conditional: "If an angle measures $30^\circ$ , then it is acute."

Name the parts first so you can move them without changing their meaning. The hypothesis p is "an angle measures 30°." The conclusion q is "it is acute."
$p \to q$
By definition, the converse swaps the hypothesis and the conclusion. This gives "If an angle is acute, then it measures 30°." You move the words "an angle" to the new hypothesis so the pronoun "it" still has a clear meaning.
$q \to p$
By definition, the inverse negates both parts and keeps them in the same order. This gives "If an angle does not measure 30°, then it is not acute."
${\sim}p \to {\sim}q$
By definition, the contrapositive swaps the parts and negates both. This gives "If an angle is not acute, then it does not measure 30°."
${\sim}q \to {\sim}p$

Answer

The converse is "If an angle is acute, then it measures $30^\circ$ ." The inverse is "If an angle does not measure $30^\circ$ , then it is not acute." The contrapositive is "If an angle is not acute, then it does not measure $30^\circ$ ."

Next, test each statement with the rule you already know: a conditional is false only when its hypothesis is true and its conclusion is false. The table applies this rule to all four statements in each of the four possible cases for $p$ and $q$ , where T means true and F means false. Apply the rule to each statement's own hypothesis and conclusion. For the contrapositive, for example, the hypothesis is ${\sim}q$ and the conclusion is ${\sim}p$ .

Truth values of all four statements in every case

p	q	Conditional: if p, then q	Converse: if q, then p	Inverse: if not p, then not q	Contrapositive: if not q, then not p
T	T	T	T	T	T
T	F	F	T	T	F
F	T	T	F	F	T
F	F	T	T	T	T

Key concept

Two pairs that always match

A conditional and its contrapositive always share a truth value, because each is false only when $p$ is true and $q$ is false. The converse and the inverse share a truth value, because each is false only when $q$ is true and $p$ is false. The converse and the inverse may differ from the original, so a true conditional does not tell you whether its converse is true. That is why proving a theorem does not prove its converse.

A statement about all angles or all polygons makes a claim about every case it covers. Until you settle that claim, treat it as a conjecture, a statement that has not been proved or disproved. A counterexample is one case where the hypothesis is true and the conclusion is false. One counterexample disproves a conjecture, and any number of confirming examples never proves it. To show that a statement is true, treat its hypothesis as given and its conclusion as what you must show, then connect them with a definition, postulate, or proved theorem.

The statements in a matching pair are false in the same row of the table, so a counterexample to one statement in a pair is also a counterexample to the other.

Worked example

Truth values when the original is true

Find the truth value of "If an angle measures $30^\circ$ , then it is acute" and of its converse, inverse, and contrapositive.

Test the original. Given: an angle measures 30°. To show: it is acute. Since 30 is greater than 0 and less than 90, the definition of an acute angle says the angle is acute. So a true hypothesis with a false conclusion cannot happen, and the conditional is true.
$0 < 30 < 90$
Use the first matching pair. A conditional and its contrapositive always share a truth value, so "If an angle is not acute, then it does not measure 30°" is true. The meaning agrees: an angle that is not acute measures 90° or more, and 30 is less than 90.
Look for a counterexample to the converse, "If an angle is acute, then it measures 30°." Take a 40° angle, a valid measure because 40 is between 0 and 180. The definition of an acute angle says it is acute, so the hypothesis is true. It does not measure 30°, so the conclusion is false. This one counterexample disproves the converse.
$0 < 40 < 90 \quad \text{and} \quad 40 \ne 30$
Use the second matching pair. The converse and the inverse share a truth value, so "If an angle does not measure 30°, then it is not acute" is false. The 40° angle is a counterexample here too: it does not measure 30°, so the hypothesis is true, and it is acute, so the conclusion is false.

Answer

The conditional and its contrapositive are true, while the converse and the inverse are false.

Worked example

Truth values when the original is false

Find the truth value of "If a polygon has four sides, then it is a square" and of its converse, inverse, and contrapositive. Use these definitions: a polygon is a closed plane figure made of straight sides, a right angle is an angle that measures exactly $90^\circ$ , and a square is a polygon with exactly four sides of equal length and four right angles.

Look for a counterexample. Take a rectangle, a polygon with exactly four sides and four right angles, whose stated side lengths are 2, 3, 2, and 3, all positive. It has four sides, so the hypothesis is true. Its sides are not all equal, so by the definition of a square it is not a square, and the conclusion is false. The conditional is false.
$2 \ne 3$
A conditional and its contrapositive always share a truth value, so "If a polygon is not a square, then it does not have four sides" is false. The same rectangle is the counterexample.
Test the converse, "If a polygon is a square, then it has four sides." Given: a polygon is a square. To show: it has four sides. The definition of a square says it has exactly four sides, so the converse is true.
The converse and the inverse share a truth value, so "If a polygon does not have four sides, then it is not a square" is true.

Answer

The conditional and its contrapositive are false, while the converse and the inverse are true.

Mistake: trusting a converse because the original is true. "If a polygon is a square, then it has four sides of equal length" is true by the definition of a square. A student reverses it to "If a polygon has four sides of equal length, then it is a square," tests squares with side lengths $1$ , $2$ , and $5$ , and calls it true. This fails because the converse may differ from the original and confirming examples never prove a statement. Testing only squares could never reveal a failure. A rhombus, a polygon with exactly four sides of equal length, can have angle measures $60^\circ$ , $120^\circ$ , $60^\circ$ , and $120^\circ$ . Its four equal sides make the hypothesis true, and with no right angles the conclusion is false. That one counterexample disproves the converse.

Sometimes a conditional and its converse are both true. Then $p$ and $q$ always go together: in every case, either both are true or both are false. A biconditional claims this in one sentence by joining $p$ and $q$ with the words "if and only if." To write one, start from a conditional and its converse, then put "if and only if" between the hypothesis and the conclusion.

p \leftrightarrow q \quad \text{means} \quad (p \to q) \text{ and } (q \to p)

A biconditional is true exactly when both directions are true. Test each direction as its own conditional, which is false only when its hypothesis is true and its conclusion is false. One counterexample to either direction makes the biconditional false. Confirming examples never prove it, so each true direction needs a definition, postulate, or proved theorem as its reason.

For example, "A polygon is a square if and only if it has four sides" is false. The direction "If a polygon is a square, then it has four sides" is true by the definition of a square. But the rectangle with side lengths $2$ , $3$ , $2$ , and $3$ disproves the direction "If a polygon has four sides, then it is a square."

A precise definition works in both directions, so you can write it as a true biconditional. For example, the definition of perpendicular lines reads, "Two lines are perpendicular if and only if they intersect to form a right angle." You can use a definition as a reason in either direction, and the next example does this with the definition of a right angle.

Worked example

Writing and evaluating a biconditional

The measure of $\angle A$ is $(2x + 10)^\circ$ . Write the conditional "If $\angle A$ is a right angle, then $x = 40$ " and its converse as one biconditional. Then decide whether the biconditional is true.

The converse swaps the two parts, which gives "If x = 40, then angle A is a right angle." By the definition of a biconditional, you join the two parts with "if and only if" to state both directions at once.
$\angle A \text{ is a right angle if and only if } x = 40$
Test the first direction. Given: angle A is a right angle. To show: x = 40. The definition of a right angle says its measure is 90°, so set the stated measure equal to 90.
$2x + 10 = 90$
Subtract 10 from both sides, using the Subtraction Property of Equality.
$2x + 10 - 10 = 90 - 10, \text{ so } 2x = 80$
Divide both sides by 2, which is not zero, using the Division Property of Equality. You reach x = 40, so this direction is true.
$\frac{2x}{2} = \frac{80}{2}, \text{ so } x = 40$
Test the converse direction. Given: x = 40. To show: angle A is a right angle. Substitute 40 for x, using the Substitution Property of Equality. The measure is 90°, which is a valid angle measure because it lies between 0° and 180°. By the definition of a right angle, angle A is a right angle, so this direction is true.
$2(40) + 10 = 80 + 10 = 90$
Both directions are true, and each one rests on a definition and properties of equality rather than on examples. So the biconditional is true.

Answer

The biconditional " $\angle A$ is a right angle if and only if $x = 40$ " is true.

Takeaway: When you rearrange a conditional, you create new claims, so check each one. A conditional and its contrapositive always share a truth value, while the converse and the inverse share a truth value and may differ from the original. A biconditional is true exactly when a conditional and its converse are both true. A conditional is false only when its hypothesis is true and its conclusion is false, so one counterexample disproves a conjecture, while any number of confirming examples never proves it.

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