converses

Converse, in logic and mathematics, refers to the reverse of a conditional statement. If a statement has the form “If P, then Q” (written P → Q), its converse is formed by swapping P and Q to yield “If Q, then P” (Q → P). The truth of the converse is not determined by the truth of the original; in general, the converse may be true or false independently.

For example, consider the original: “If a figure is a rectangle, it has four sides.” The converse would be: “If a figure has four sides, it is a rectangle.” This converse is false in general, since many four-sided figures are not rectangles (such as trapezoids or rhombuses). Conversely, some statements have true converses, but the truth cannot be inferred from the original alone. The contrapositive of the original, “If not Q, then not P,” is logically equivalent to the original, while the inverse, “If not P, then not Q,” is not automatically equivalent.

Beyond conditionals, the term converse appears in relation theory. The converse (or inverse) of a relation R is the relation R⁻¹, obtained by swapping the order of the pairs: (a, b) ∈ R implies (b, a) ∈ R⁻¹. A common example is the relation “is parent of”; its converse is “has child.”

In practice, distinguishing a statement from its converse matters in proofs, programming, and data reasoning. Proving the original does not by itself establish the truth of the converse, and additional justification is often required. See also contrapositive, inverse, and relation.