Question 123503
Given a conditional statement p --> q, find the inverse of its inverse, the inverse of its converse, and the inverse of its contrapositive.
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To find the converse, swap the left and right sides of <font face = "symbol">®</font>
To find the inverse, negate left and right sides of <font face = "symbol">®</font>
To find the contrapositive, swap and negate left and right sides of <font face = "symbol">®</font>

To find the inverse of the inverse of p <font face = "symbol">®</font> q

1. First find the inverse of p <font face = "symbol">®</font> q by negating both sides, getting
   ~p <font face = "symbol">®</font> ~q
2. Second, find the inverse of that by negating both sides, getting
   ~~p <font face = "symbol">®</font> ~~q which simplifies to 
   p <font face = "symbol">®</font> q.  So the inverse of the inverse is the original statement.

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To find the inverse of the contrapositive of p <font face = "symbol">®</font> q

1. First find the contrapositive of p <font face = "symbol">®</font> q by swapping and negating
   the left and right sides, getting
   ~q <font face = "symbol">®</font> ~p
2. Second, find the inverse of that by negating both sides, getting
   ~~q <font face = "symbol">®</font> ~~p which simplifies to q <font face = "symbol">®</font> p.  So the inverse of the 
   contrapositive is the converse (of the original statement).

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Incidentally, given an original statement:

1. The inverse of the inverse is the original statement.
2. The inverse of the converse is the contrapositive.
3. The inverse of the contrapositive is the converse.
4. The converse of the inverse is the contrapositive.
5. The converse of the converse is the original statement.
6. The converse of the contrapositive is the inverse.
7. The contrapositive of the inverse is the converse.
8. The contrapositive of the converse is the inverse.
9. The contrapositive of the contrapositive is the original statement.

Edwin</pre>