Converse, Inverse, and Contrapositive

After learning how to negate an implication, we next compare the related conditional statements formed by reversing or negating its parts. The implication, converse, inverse, and contrapositive are often confused in ordinary language. In mathematical logic, each has a precise symbolic form. In this lesson, converse, inverse, and contrapositive will be distinguished by logical equivalence and truth values.

Let

p\to q

be an implication. The \textbf{contrapositive} of

p\to q

\neg q\to\neg p

Let

p\to q

be an implication. The \textbf{contrapositive} of

p\to q

\neg q\to\neg p

Let

p\to q

be an implication. The \textbf{converse} of

p\to q

q\to p

Let

p\to q

be an implication. The \textbf{inverse} of

p\to q

\neg p\to\neg q

Use the sorter to keep the four related conditional forms separate. The original implication keeps the order

p

then

q

, the converse reverses the order, the inverse negates both parts, and the contrapositive both reverses and negates. The sorter also marks which forms are logically equivalent. This helps prevent the common mistake of treating a statement and its converse as the same claim.

Let

p

and

q

be primitive statements. Then

p\to q\Leftrightarrow\neg q\to\neg p

Given that

p

and

q

are primitive statements. To prove that

p\to q\Leftrightarrow\neg q\to\neg p

. Using implication as disjunction,

\begin{equation*} p\to q\Leftrightarrow\neg p\lor q. \end{equation*}

Also,

\begin{align*} \neg q\to\neg p&\Leftrightarrow\neg(\neg q)\lor\neg p\\ &\Leftrightarrow q\lor\neg p\\ &\Leftrightarrow\neg p\lor q. \end{align*}

Therefore, both statements are equivalent to

\neg p\lor q

. Hence,

p\to q\Leftrightarrow\neg q\to\neg p

\square

Let

p

and

q

be primitive statements. Then

q\to p\Leftrightarrow\neg p\to\neg q

Given that

p

and

q

are primitive statements. To prove that

q\to p\Leftrightarrow\neg p\to\neg q

. The contrapositive of

q\to p

\neg p\to\neg q

. Since an implication is logically equivalent to its contrapositive,

\begin{equation*} q\to p\Leftrightarrow\neg p\to\neg q. \end{equation*}

Hence, the converse and inverse of an implication are logically equivalent to each other.

\square

In general,

p\to q\not\Leftrightarrow q\to p

. The truth of an implication does not by itself guarantee the truth of its converse.

In general,

\neg p\to\neg q\not\Leftrightarrow\neg q\to\neg p

. The inverse is equivalent to the converse, not to the original implication.

Let

p

mean Today is Thanksgiving, and let

q

mean Tomorrow is Friday. The implication

p\to q

says: If today is Thanksgiving, then tomorrow is Friday. The contrapositive

\neg q\to\neg p

says: If tomorrow is not Friday, then today is not Thanksgiving. The converse

q\to p

says: If tomorrow is Friday, then today is Thanksgiving. The inverse

\neg p\to\neg q

says: If today is not Thanksgiving, then tomorrow is not Friday.

Show that an implication is not logically equivalent to its converse.

Let

p\to q

be an implication. Suppose

p=0

and

q=1

. Then

p\to q=1

, but

q\to p=0

. Since there is at least one truth assignment for which the two statements have different truth values,

p\to q\not\Leftrightarrow q\to p

. Hence, an implication is not logically equivalent to its converse.

Translate necessary and sufficient conditions using converse and contrapositive.

The statement

p

is sufficient for

q

means

p\to q

. The statement

p

is necessary for

q

means

q\to p

. Therefore, the converse of

p

is sufficient for

q

p

is necessary for

q

. Since

p\to q\Leftrightarrow\neg q\to\neg p

, necessary and sufficient conditions can also be studied through contrapositives.

For the implication If

n

is divisible by

4

, then

n

is even, write the converse, inverse, and contrapositive.

Converse: If

n

is even, then

n

is divisible by

4

. Inverse: If

n

is not divisible by

4

, then

n

is not even. Contrapositive: If

n

is not even, then

n

is not divisible by

4

An implication is logically equivalent to its contrapositive.

No. The converse may be true or false, but it is not guaranteed by the truth of the original implication.

The inverse is logically equivalent to the converse.

BMLabs Mathematics

BMLabs Mathematics

Converse, Inverse, and Contrapositive

Converse, Inverse, and Contrapositive

CONTRAPOSITIVE

CONVERSE

INVERSE

SORTING CONDITIONAL FORMS

CONDITIONAL FORM SORTER

CONTRAPOSITIVE EQUIVALENCE

CONVERSE-INVERSE EQUIVALENCE

NON-EQUIVALENCE OF IMPLICATION AND CONVERSE

NON-EQUIVALENCE OF INVERSE AND CONTRAPOSITIVE

THANKSGIVING CONDITIONAL

FREQUENTLY ASKED QUESTIONS

COMPARE CONDITIONAL FORMS

Introduction To Discrete Mathematics

Paths and Circuits