Definition:Converse Statement

Definition

The converse of the conditional:

$p \implies q$

is the statement:

$q \implies p$

Examples

$x < 5$ and $x \le 5$

Let:

$P$ be the statement $x < 5 \implies x \le 5$

$Q$ be the statement $x \le 5 \implies x < 5$

$R$ be the statement $x > 5 \implies x \ge 5$

$S$ be the statement $x \ge 5 \implies x > 5$

for $x \in \R$.

Then:

$P$ and $Q$ are converse statements

$R$ and $S$ are converse statements

$P$ and $R$ are contrapositive statements

$Q$ and $S$ are contrapositive statements

$P$ and $R$ are true

$Q$ and $S$ are false.

Also see

• Conditional and Converse are not Equivalent: The converse of a true conditional is not necessarily true, and the converse of a false conditional is not necessarily false.

• Definition:Inverse Statement
• Definition:Contrapositive Statement

Linguistic Note

The word converse, in the context of the term converse statement, is pronounced with the stress on the first syllable: con-verse.

When pronounced con-verse, with the stress on the second syllable, it means to communicate (usually verbally) with others.

Sources

• 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S \text{II}.10$: Equivalence of sentences
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): Chapter $1$: Sets, Functions, and Relations: $\S 2$: Some Remarks on the Use of the Connectives and, or, implies
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Axioms
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 4$: The implies sign ($\Rightarrow$)
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): converse (of a theorem)
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): converse (of a theorem)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): converse