Editing Converse (logic) (section)

===Converse of a theorem===
In mathematics, the converse of a theorem of the form ''P'' → ''Q'' will be ''Q'' → ''P''. The converse may or may not be true, and even if true, the proof may be difficult. For example, the [[four-vertex theorem]] was proved in 1912, but its converse was proved only in 1997.<ref>{{Cite web|url=https://www.math.colostate.edu/~clayton/research/talks/FourVertexPrint.pdf|title=The Four Vertex Theorem and its Converse|last=Shonkwiler|first=Clay|date=October 6, 2006|website=math.colostate.edu|access-date=2019-11-26}}</ref>

In practice, when determining the converse of a mathematical theorem, aspects of the antecedent may be taken as establishing context. That is, the converse of "Given P, if Q then R''"'' will be "Given P, if R then Q''"''. For example, the [[Pythagorean theorem]] can be stated as:

<blockquote>
''Given'' a triangle with sides of length ''<math>a</math>'', ''<math>b</math>'', and ''<math>c</math>'', ''if'' the angle opposite the side of length ''<math>c</math>'' is a right angle, ''then'' <math>a^2 + b^2 = c^2</math>'''.'''
</blockquote>

The converse, which also appears in [[Euclid's Elements|Euclid's ''Elements'']] (Book I, Proposition 48), can be stated as:

<blockquote>
''Given'' a triangle with sides of length ''<math>a</math>'', ''<math>b</math>'', and ''<math>c</math>'', ''if'' <math>a^2 + b^2 = c^2</math>, ''then'' the angle opposite the side of length ''<math>c</math>'' is a right angle.
</blockquote>