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.
~~~~~~~~~~~~~~~~~~~~~
I will solve (ii) and (iii).
(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.
Solved.
-----------------
The lesson to learn is this:
To prove that the operation is associative, the proof should be done for all x, y, z.
To prove that the operation is not associative, it is enough to show it using special counter-examples.