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Version 10 |
| \PMlinkescapeword{congruent} |
\PMlinkescapeword{congruent} |
| \PMlinkescapeword{contains} |
\PMlinkescapeword{contains} |
| \PMlinkescapeword{words} |
\PMlinkescapeword{words} |
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| Let a statement be of the form of an implication |
Let a statement be of the form of an implication |
| $$\mbox{If } p, \mbox{ then } q$$ |
$$\mbox{If } p, \mbox{ then } q$$ |
| \PMlinkname{i.e.}{Ie} it has a certain premise $p$ and a conclusion $q$. The statement in which one has interchanged the conclusion and the premise, |
\PMlinkname{i.e.}{Ie} it has a certain premise $p$ and a conclusion $q$. The statement in which one has interchanged the conclusion and the premise, |
| $$\mbox{If } q, \mbox{ then } p$$ |
$$\mbox{If } q, \mbox{ then } p$$ |
| is the {\em converse} of the first. In other words, from the former one concludes that $q$ is necessary for $p$, and from the latter that $p$ is necessary for $q$. |
is the {\em converse} of the first. In other words, from the former one concludes that $q$ is necessary for $p$, and from the latter that $p$ is necessary for $q$. |
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| Note that the converse of an implication and the inverse of the same implication are contrapositives of each other and thus are logically equivalent. |
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| If the original statement is a theorem that is known to be true, then its converse is the \emph{converse theorem} of the original statement. Note that, if the converse theorem of a true theorem ``If $p$, then $q$'' is also true, then ``$p$ iff $q$'' is a true theorem. \\ |
If the original statement is a theorem that is known to be true, then its converse is the \emph{converse theorem} of the original statement. Note that, if the converse theorem of a true theorem ``If $p$, then $q$'' is also true, then ``$p$ iff $q$'' is a true theorem. \\ |
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| For example, we know the theorem on isosceles triangles: |
For example, we know the theorem on isosceles triangles: |
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| {\em If a triangle contains two \PMlinkname{congruent}{Congruent2} sides, then it has two congruent angles.} |
{\em If a triangle contains two \PMlinkname{congruent}{Congruent2} sides, then it has two congruent angles.} |
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| There is also its converse theorem: |
There is also its converse theorem: |
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| {\em If a triangle contains two congruent angles, then it has two congruent sides.} |
{\em If a triangle contains two congruent angles, then it has two congruent sides.} |
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| Both of these theorems are true (see the entries angles of an isosceles triangle and determining from angles that a triangle is isosceles, respectively). But there are many true theorems whose converse theorem is not true, \PMlinkname{e.g.}{Eg}: |
Both of these theorems are true (see the entries angles of an isosceles triangle and determining from angles that a triangle is isosceles, respectively). But there are many true theorems whose converse theorem is not true, \PMlinkname{e.g.}{Eg}: |
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| {\em If a function is differentiable on an interval $I$, then it is continuous on $I$.} |
{\em If a function is differentiable on an interval $I$, then it is continuous on $I$.} |
