Question 1203441
Which of the ff are associative binary operations? Show your solutions? 
i. (ℤ,*), where x*y=(x+y)-(x·y) for all x,y,∈ℤ. 
ii. (ℝ,*), where x*y=max (x,y) for all x,y,∈ℝ. 
iii. (ℝ,*), where x*y= |x+y|  for all x,y,∈ℝ.
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                I will solve  (ii)  and  (iii).



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(ii)   ((x*y)*z) = (max(x,y)*z) = obviously = max(x,y,z).

       (x*(y,z)) = (x*max(y,z)) = obviously = max(x,y,z).


       The expressions generate the same output for all x, y, z, so this operation (*) is associative.



(iii) Take  x= 1,  y= 2, z= -3.  Then

      ((x*y)*z) = ||x+y| + z| = ||1+2| + (-3)| = |3 + (-3)| = |0| = 0.

      (x*(y*z)) = |x + |y+z|| = |1 + |2+(-3)|| = |1 + |-1|| = |1 + 1| = |2| = 2.


       The expressions generate different results for given x, y, z, so this operation (*) is NOT associative.
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Solved.


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<H3>The lesson to learn is this:</H3>

    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;To prove that the operation is associative, the proof should be done for all x, y, z.    

    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;To prove that the operation is not associative, it is enough to show it using special counter-examples.
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