# Prove commutative and associative

## Homework Statement

a function 'd' is a closed binary operation on a set called 'T'.
There is an identity element named j.
for all elements a, b, and c in the set 'T', we have d(a, d(b,c)) = d((a,c), b)

can anyone help me show that d is commutative and associative?

## Homework Equations

f is commutative if function f(s,u) = f(u,s)
f is associative if function f(x, f(y,z)) = f((x,y), z)
q is an identity element if function f(q,a) = a and f(a,q) = a.

## The Attempt at a Solution

The attempt began with proving commutative, but ended shortly after because there are three elements in the problem and as far as i know commutative only uses 2.

Dick
Homework Helper
Try putting one of your three variables to j.

epenguin
Homework Helper
Gold Member
May show up my unfamiliarity :shy: but have you quoted the associativity law right?

Likewise in the first line d((a,c), b) ? You have got this (a,c) in the air with no operation on the pair or rule to do anything with them, if I am not mistaken. :shy:

yes the law is correct. What was given to me in the problem is misleading regarding the law of associativity which is why i am so stumped. as for j, I am not sure what putting one of my variables in for j will do, but i will mess with it and see where it gets me. Thanks!

epenguin
Homework Helper
Gold Member
Then I am the one who needs help.

Associativity is written x*(y*z) = (x*y)*z where * is an operation.

Translating this into the f language, this is

f(x, f(y,z)) = f((f(x,y),z) .

Both are formulae saying combine two things then combine the result with this other thing can be done in this sequence, but if you do it in this other sequence the result is the same. But it seems to me that in your formula you also have these brackets e.g. (x,y) with no defined operation to combine them so they do not mean anything definite or defined?

Dick
Homework Helper
Then I am the one who needs help.

Associativity is written x*(y*z) = (x*y)*z where * is an operation.

Translating this into the f language, this is

f(x, f(y,z)) = f((f(x,y),z) .

Both are formulae saying combine two things then combine the result with this other thing can be done in this sequence, but if you do it in this other sequence the result is the same. But it seems to me that in your formula you also have these brackets e.g. (x,y) with no defined operation to combine them so they do not mean anything definite or defined?

Yes, there's a 'd' missing in the given relation and an 'f' missing in the statement of associativity. I figured they were just typos.

epenguin
Homework Helper
Gold Member
I am relieved!

That should make it easier. Deveno

## Homework Statement

a function 'd' is a closed binary operation on a set called 'T'.
There is an identity element named j.
for all elements a, b, and c in the set 'T', we have d(a, d(b,c)) = d((a,c), b)

can anyone help me show that d is commutative and associative?

## Homework Equations

f is commutative if function f(s,u) = f(u,s)
f is associative if function f(x, f(y,z)) = f((x,y), z)
q is an identity element if function f(q,a) = a and f(a,q) = a.

## The Attempt at a Solution

The attempt began with proving commutative, but ended shortly after because there are three elements in the problem and as far as i know commutative only uses 2.

this is NOT associativity, as stated. if we write d(a,b) as a*b, what he is given is:

a*(b*c) = (a*c)*b (note the reversal of b and c).

it is far easier to prove commutivity FIRST.

this is what the hint on using j is meant to convey.