Answer to Question #102206 in Abstract Algebra for Kampamba

Question #102206
Given that x*y=1+x+y, show wheather the binary operation is associative algebrically
1
Expert's answer
2020-02-04T08:40:25-0500

A binary operation "\\cdot" is associative when "(x \\cdot y) \\cdot z = x \\cdot (y \\cdot z)" for any "x, y, z" .


Suppose we have

"(x \\cdot y) \\cdot z"

We can replace the binary operation with addition using its definition:

"(x \\cdot y) \\cdot z = (1 + x + y) \\cdot z = 1 + (1+x+y) +z"


Now we can remove parentheses (using associativity of addition), regroup terms (using commutativity of addition) and put parentheses where we want (using associativity of addition once again):

"1 + (1 + x + y) + z = \\\\\n=1 + 1 + x + y + z = \\\\\n= 1 + x + 1+y +z = \\\\\n=1 + x + (1+y+z)"

"1+y+z = y \\cdot z" by definition of this binary operation.

"1+ x + (1+y+ z) = 1+x+(y \\cdot z)"

"1+x+(y \\cdot z) = x \\cdot (y \\cdot z)" by definition of our binary operation.


We proved that "(x \\cdot y) \\cdot z = x\\cdot (y \\cdot z)" which means we proved that the binary operation is associative algebraically. What was to be shown.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS