Editing Counterexample (section)

===Rectangle example===
Suppose that a mathematician is studying [[geometry]] and [[shape]]s, and she wishes to prove certain theorems about them. She [[conjecture]]s that "All [[rectangles]] are [[Square (geometry)|squares]]", and she is interested in knowing whether this statement is true or false. 

In this case, she can either attempt to [[Mathematical proof|prove]] the truth of the statement using [[deductive reasoning]], or she can attempt to find a counterexample of the statement if she suspects it to be false. In the latter case, a counterexample would be a rectangle that is not a square, such as a rectangle with two sides of length 5 and two sides of length 7. However, despite having found rectangles that were not squares, all the rectangles she did find had four sides. She then makes the new conjecture "All rectangles have four sides". This is logically weaker than her original conjecture, since every square has four sides, but not every four-sided shape is a square.

The above example explained — in a simplified way — how a mathematician might weaken her conjecture in the face of counterexamples, but counterexamples can also be used to demonstrate the necessity of certain assumptions and [[hypothesis]]. For example, suppose that after a while, the mathematician above settled on the new conjecture "All shapes that are rectangles and have four sides of equal length are squares". This conjecture has two parts to the hypothesis: the shape must be 'a rectangle' and must have  'four sides of equal length'. The mathematician then would like to know if she can remove either assumption, and still maintain the truth of her conjecture. This means that she needs to check the truth of the following two statements:

# "All shapes that are rectangles are squares."
# "All shapes that have four sides of equal length are squares".

A counterexample to (1) was already given above, and a counterexample to (2) is a non-square [[rhombus]]. Thus, the mathematician now knows that each assumption by itself is insufficient.